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+Bipol: Multi-axes Evaluation of Bias with
+Explainability in Benchmark Datasets
+Tosin Adewumi∗‡, Isabella S¨odergren†, Lama Alkhaled‡, Sana Sabah Sabry‡, Foteini Liwicki‡
+and Marcus Liwicki‡
+Machine Learning Group, EISLAB,
+Lule˚a University of Technology, Sweden
+∗corresponding author, †isasde-5@student.ltu.se, ‡firstname.lastname@ltu.se
+Abstract—We evaluate five English NLP benchmark datasets
+(available on the superGLUE leaderboard) for bias, along mul-
+tiple axes. The datasets are the following: Boolean Question
+(Boolq), CommitmentBank (CB), Winograd Schema Challenge
+(WSC), Winogender diagnostic (AXg), and Recognising Textual
+Entailment (RTE). Bias can be harmful and it is known to be
+common in data, which ML models learn from. In order to
+mitigate bias in data, it is crucial to be able to estimate it
+objectively. We use bipol, a novel multi-axes bias metric with
+explainability, to quantify and explain how much bias exists in
+these datasets. Multilingual, multi-axes bias evaluation is not very
+common. Hence, we also contribute a new, large labelled Swedish
+bias-detection dataset, with about 2 million samples; translated
+from the English version. In addition, we contribute new multi-
+axes lexica for bias detection in Swedish. We train a SotA model
+on the new dataset for bias detection. We make the codes, model,
+and new dataset publicly available.
+Index Terms—bias, explainability, bipol, dataset, nlp
+I. INTRODUCTION
+Bias, which can be harmful [1], is the unfair prejudice in
+favor of or against a thing, person or group, relative to another
+[2]. Measuring bias in text data can be challenging because of
+the axes that may be involved (e.g. religious or gender bias).
+Bipol was introduced by [3]. It is a metric that estimates bias
+along multiple axes in text data and provides an explanation
+for its scores.
+In this work, we investigate and estimate social bias in some
+of the benchmark datasets for NLP, particularly those available
+on the English SuperGLUE leaderboard. The SuperGLUE
+was introduced by [4] and provides benchmark datasets for
+different NLP tasks. Benchmark datasets are datasets for
+comparing the performance of algorithms for specific use-
+cases. [5], [6]. Such datasets have been the foundation for
+some of the significant advancements in the field [6]. We
+investigate the following datasets: Boolq, CB, WSC, AXg, and
+RTE.
+Classification accuracy is known to drop with attempts at
+mitigating biases in data [7]–[9] yet it is important to estimate
+and mitigate them because of the ethical implications or harm
+that may arise for the disadvantaged, sensitive group [10], [11],
+thereby affecting the data quality. Some characteristics of bias
+in text data are:1
+• It is heavily lopsided.
+1https://libguides.uwgb.edu/bias
+• It uses inappropriate language.
+• It is based on unsubstantiated claims.
+a) Our
+contributions:
+We
+show
+quantitatively
+and
+through explainability that bias exists in the datasets. This
+will provide researchers with insight into how to mitigate
+bias in text data and possibly add impetus to the conversation
+on whether it is even ethical to remove these social biases
+from the training data, because they represent the real world.
+Furthermore, we provide, possibly, the largest labelled dataset
+and lexica for bias detection in Swedish (multi-axes bias
+dataset (MAB)-Swedish) and train a model based on the state-
+of-the-art (SotA) Swedish BERT [12]. We release our codes
+publicly.2
+The rest of this paper is structured as follows: Section II
+describes materials used and our methods, including details of
+the characteristics of bipol and the new MAB-Swedish dataset.
+Section III describes the results and discusses the types of bias
+in the datasets. Section IV discusses some of the previous
+related work. In Section V, we conclude our work.
+II. MATERIALS & METHODS
+A. Bipol
+There are two stages in the implementation of bipol (see
+1a [3]) before it gives a final score between 0.0 (zero or un-
+detected bias) and 1.0 (extreme bias). The first stage involves
+the classification of the data samples (into biased and unbiased
+categories) using a trained model (see 1b). It is the ratio of
+the number of biased samples (true positives (tp) and false
+positives (fp)) to the total samples (true positives (tp), false
+positives (fp), true negatives (tn), and false negatives (fn)).
+Ideally, a good classifier should minimize the number of fp
+and maximize the number of tp.
+The second stage evaluates the biased samples for sensitive
+terms listed in the multi-axes lexica (see 1c). It involves finding
+the difference between the two maximum summed frequencies
+in the types of an axis (| �n
+s=1 as − �m
+s=1 bs|), which is then
+divided by the summed frequencies of all the terms in that axis
+(�p
+s=1 ds). The average over all the axes ( 1
+q
+�q
+x=1) using this
+operation is then averaged over all the biased samples ( 1
+r
+�r
+t=1
+). Table I provides the Swedish lexica sizes. The lexica are
+2github.com/tosingithub/Bipol
+arXiv:2301.12139v1  [cs.CL]  28 Jan 2023
+
+derived from [13], [14] and Wikipedia3 and may be expanded
+as needed. The English lexica contain more and are derived
+from public sources [3].
+b = bc.bs
+(1a)
+bc =
+tp + fp
+tp + fp + tn + fn
+(1b)
+bs = 1
+r
+r
+�
+t=1
+�
+1
+q
+q
+�
+x=1
+�| �n
+s=1 as − �m
+s=1 bs|
+�p
+s=1 ds
+�
+x
+�
+t
+(1c)
+TABLE I
+SWEDISH LEXICA SIZES. THESE MAY BE EXPANDED.
+Axis
+Axis type 1
+Axis type 2
+Gender
+17 (female)
+19 (male)
+Racial
+10 (black)
+10 (white)
+The rationale for using bipol is because of the strengths
+of the metric. These include 1) the relative simplicity of
+calculating a score, 2) it is straight-forward to implement since
+it is based on existing concepts like lexica and classifiers, 3)
+it captures semantic and term frequency (TF) aspects of data,
+and 4) it is not limited in the total number of axes that may
+be used. We acknowledge, however, that it has limitations that
+are based on the limitations of the tools that may be used to
+calculate it.
+B. Datasets
+The new MAB-Swedish dataset: The dataset was machine-
+translated (from MAB [3]) using the Helsinki-NLP model
+[15], which was mostly trained with guided alignment. The
+automatic translation took over 48 hours on one GPU. The
+features in the two datasets are, hence, the same. It has
+1,946,975 samples, as given in Table II. The English version
+was constructed from two datasets: Jigsaw4 and the Social
+Bias Inference Corpus v2 (SBICv2) by [16]. The Jigsaw
+(a multipurpose dataset) and the SBICv2 assume that bias
+correlates with toxicity. The samples with toxicity equal to
+or above the threshold of 0.1 were automatically labelled
+as biased and those below were labelled as unbiased. Some
+examples in the MAB-Swedish are given in Table III.
+TABLE II
+MAB-SWEDISH & MAB DATASETS SPLIT
+Set
+Biased
+Unbiased
+Total
+Training
+533,544
+1,209,433
+1,742,977
+Validation
+32338
+69649
+101987
+Test
+33470
+68541
+102011
+599,352
+1,347,623
+1,946,975
+3en.wikipedia.org/wiki/Swedish profanity
+4medium.com/jigsaw/creating-labeled-datasets-and-exploring-the-role-of-
+human-raters-56367b6db298
+TABLE III
+EXAMPLES FROM THE MAB-SWEDISH (THE ENGLISH IN THE ORIGINAL
+IS IN ITALICS)
+comment text
+label
+old id
+id
+Den h¨ar slynan ¨ar galen som
+skulle
+l¨asa
+en
+bok
+av
+en
+kvinna.
+biased
+239612
+1212584
+(This b*tch is nuts. Who would
+read a book by a woman.)
+9 av 10 m¨anniskor njuter av
+mobbning. Jag kommer alltid
+att vara den 10: e
+biased
+none
+1517517
+(9 out of 10 people enjoy bully-
+ing. I will always be the 10th.)
+”F¨orfattaren s¨ager att det finns
+¿ m˚anga, m˚anga fler anled-
+ningar till att jag antar att en
+uppf¨oljare finns i verken, d˚a?”
+unbiased
+383811
+110831
+(”The author says that there
+are many, many more reasons I
+am assuming a sequel is in the
+works, then?”)
+Vad kallar du underkl¨ader f¨or
+araber?
+biased
+none
+1618146
+(What do you call lingerie for
+Arabs? Socks.)
+C. Experiments
+The experiments are conducted on two shared Nvidia DGX-
+1 clusters running Ubuntu 18.04 and 20.04 with 8 × 32GB
+V100 and 8 x 40GB A100 GPUs, respectively. Average results
+are reported after running each experiment twice. To evaluate
+the benchmark datasets, we use RoBERTa, DeBERTa, and
+Electra bias-detection trained models [3].
+Wandb [17], an experiment tracking tool, is run for 5 counts
+for training Swedish BERT with bayesian optimization to
+suggest the best hyper-parameter combination for the initial
+learning rate (1e-3 - 2e-5) and epochs (6 - 10), since it has
+been observed that hyper-parameters strongly influence per-
+formance [18]. Figure 13 (in the appendix) shows the wandb
+exploration for Swedish BERT on MAB-Swedish in parallel
+coordinates. We use the pretrained base Swedish BERT [12]
+from the HuggingFace hub [19]. Average training time was 15
+hours. Average evaluation time ranges from about 30 minutes
+to over 24 hours for the English benchmark datasets.5
+III. RESULTS AND DISCUSSION
+The macro F1 score on the validation set of MAB-Swedish
+is 0.8688 and standard deviation (s.d.) of 0.0005 (see 13).
+From Table IV we observe that all the datasets have bias,
+though little. The dataset with the least amount of bias is
+Boolq, which is confirmed by all the three models. This
+is despite the dataset having the highest number of unique
+samples. CB has the largest amount of bias and this is also
+confirmed by the three models.
+5particularly when cpulimit is used, in fairness to other users
+
+TABLE IV
+RESULTS OF AVERAGE SCORES.
+bipol level ↓ (s.d.)
+RoBERTa
+unique samples
+corpus
+sentence
+bipol (b)
+Boolq
+7,929
+0.0066
+0.8027
+0.0053 (0)
+CB
+250
+0.08
+0.8483
+0.0679 (0)
+WSC
+279
+0.0466
+0.8718
+0.0406 (0)
+AXg
+178
+0.0112
+1
+0.0112 (0)
+RTE
+2,379
+0.0294
+0.8518
+0.0251 (0)
+DeBERTa
+Boolq
+7,929
+0.0103
+0.7212
+0.0075 (0)
+CB
+250
+0.084
+0.9048
+0.076 (0)
+WSC
+279
+0.0609
+1
+0.0609 (0)
+AXg
+178
+0.0112
+1
+0.0112 (0)
+RTE
+2,379
+0.0366
+0.8655
+0.0316 (0)
+Electra
+Boolq
+7,929
+0.0073
+0.8089
+0.0059 (0)
+CB
+250
+0.0316
+0.881
+0.074 (0)
+WSC
+279
+0.0609
+0.9559
+0.0582 (0)
+AXg
+178
+0.0112
+1
+0.0112 (0)
+RTE
+2,379
+0.0269
+0.8593
+0.0231 (0)
+Explaining bias type
+The type of overall bias (for the gender axis) in many of
+the datasets is explained by the dictionary of lists produced
+by bipol (see Appendix B) and represented in ”top-5 frequent
+terms” bar graphs of Figures 1 to 12. We observe from Figures
+1, 2, and 3 that Boolq is male-biased. Figures 4, 5, and 6 show
+that CB is also male-biased. This is the case also for RTE, as
+revealed by Figures 7, 8, and 9. On the other hand, we observe
+that the case of WSC is not clear-cut because Figure 10 shows
+only a marginal lead for female bias, Figure 11 shows the
+difference among the top-5 is zero and Figure 12 shows a
+slight overall male bias.
+Fig. 1. Top-5 gender frequent terms in Boolq by RoBERTa.
+IV. RELATED WORK
+Bias can lead to unfair treatment based on factors such as
+gender, age, race, etc [3]. Determining the level of bias in
+NLP datasets along these multiple axes can be a significant
+challenge but there has been considerable effort in identifying
+and analyzing bias along some of these axes [20]–[23]. Studies
+have demonstrated that the biases in language models for the
+intersection of gender and race can be greater than those for
+Fig. 2. Top-5 gender frequent terms in Boolq by DeBERTa.
+Fig. 3. Top-5 gender frequent terms in Boolq by Electra.
+Fig. 4. Top-5 gender frequent terms in CB by Roberta.
+Fig. 5. Top-5 gender frequent terms in CB by DeBERTa.
+Fig. 6. Top-5 gender frequent terms in CB by Electra.
+
+90
+80
+80
+70
+60
+49
+50
+40
+30
+23
+17
+20
+10
+6
+8
+5
+10
+3
+3
+0
+hel she
+him I her
+male I female
+boy I wife
+jackl love
+Term
+Male
+Female100
+93
+89
+81
+77
+80
+Frequency
+60
+40
+20
+13
+11
+10
+9
+3 
+4
+0
+hel she
+him I her
+malel love
+jackI female
+guyl woman
+Term
+Male
+Female80
+70
+70
+60
+60
+51
+Frequency
+50
+43
+40
+30
+20
+10
+7
+8
+6
+10
+3
+2
+0
+hel she
+him I her
+malel love
+jackl female
+guyl woman
+Male
+Female20
+17
+15
+Frequency
+10
+8
+7
+5
+1
+0
+0
+0
+0
+0
+0
+hel she
+him I her
+boy I girl
+manI woman
+male I female
+Term
+Male
+Female20
+16
+15
+10
+Frequency
+5
+4
+5
+1
+1
+0
+0
+0
+0
+0
+0
+hel she
+him I her
+boyI girl
+manI woman
+maleI female
+Term
+IMale
+Female16
+14
+14
+12
+Frequency
+10
+8
+6
+6
+4
+4
+1
+2
+1
+0
+0
+0
+0
+0
+0
+hel she
+him I her
+boyl girl
+manI woman
+malel female
+Term
+Male
+FemaleFig. 7. Top-5 gender frequent terms in RTE by RoBERTa.
+Fig. 8. Top-5 gender frequent terms in RTE by DeBERTa.
+Fig. 9. Top-5 gender frequent terms in RTE by Electra.
+Fig. 10. Top-5 gender frequent terms in WSC by RoBERTa.
+Fig. 11. Top-5 gender frequent terms in WSC by DeBERTa.
+Fig. 12. Top-5 gender frequent terms in WSC by Electra.
+gender and race individually and that addressing bias along
+only one axis can lead to more issues [24], [25]. Our work
+does not limit the number of axes that can be evaluated.
+Addressing bias in the English language is not sufficient.
+[26] proposed a multi-language approach using HurtLex [27].
+In the English language, there are common biases that asso-
+ciate female terms with subjects such as liberal arts and family
+and male terms with subjects such as science [28]. There are
+also more words that sexualize females more than males [22].
+Other languages have their own peculiarities.
+There are various methods for quantifying the extent of
+discrimination or bias that is present in a dataset. One method
+is odds ratio (OR), which compares the chance of a specific
+outcome happening, with a certain exposure, to the likelihood
+of that outcome happening without the exposure [29]. Another
+method is the impact ratio (IR), which calculates the ratio of
+positive outcomes for a protected group to the general group.
+In [20], they compare lexicon method to model classification.
+Our approach combines the strengths of both approaches.
+Other researchers have quantitatively shown the bias present
+in the geometry of word embeddings, which may amplify
+different gender or demographic stereotypes [30]–[32]. To
+address the bias in word embeddings, [33] suggests debiasing
+by removing gender from the embeddings.
+V. CONCLUSION
+We show that all benchmark datasets we evaluated, which
+are available on the SuperGLUE leaderboard, contain bias
+to different degrees. This is the first time these datasets are
+evaluated in such a way that quantifies the amount of bias and
+the type. We believe these evaluations will motivate research
+on how to more effectively mitigate bias along multiple axes in
+datasets. Our public release of the new MAB-Swedish dataset,
+lexica and model will also facilitate future work in multilingual
+bias detection.
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+APPENDIX
+A. Data
+1) BoolQ (Boolean Questions): is a question-answering
+(QA) task where each example has a short passage and a
+
+yes/no question about the passage [34] . These questions were
+provided anonymously by Google search users and afterwards
+paired with a paragraph from a Wikipedia article that has the
+answer. We evaluated the passage column of the dataset.
+2) CB:
+[35]: contains short texts in which at least one
+sentence has an embedded clause. The resulting task is framed
+as three-class textual entailment on examples that are drawn
+from the following datasets: Wall Street Journal, fiction from
+the British National Corpus, and Switchboard. We evaluated
+the premise column of the dataset.
+3) WSC:
+[36]: is a coreference resolution dataset. Exam-
+ples consist of a sentence with a pronoun and a list of noun
+phrases from the sentence. We evaluated the text column of
+the dataset.
+4) AXg:
+[37]: It is designed to measure gender bias in
+coreference resolution systems. Each example consists of a
+premise sentence having a male or female pronoun and a
+hypothesis giving a possible antecedent of the pronoun. We
+evaluated the premise column of the dataset.
+5) RTE [4]: : datasets come from a series of annual compe-
+titions on textual entailment. Data from several sources were
+merged and converted to two-class classification: entailment
+and not entailment to obtain this dataset. We evaluated the
+premise column of the dataset.
+B. Experiment
+Dictionary of lists for RoBERTa on Boolq: {’gender’: [’
+she ’: 23, ’ her ’: 17, ’ woman ’: 2, ’ lady ’: 1, ’ female ’:
+6, ’ girl ’: 1, ’ skirt ’: 0, ’ madam ’: 0, ’ gentlewoman ’: 0,
+’ madame ’: 0, ’ dame ’: 0, ’ gal ’: 0, ’ maiden ’: 0, ’ maid
+’: 0, ’ damsel ’: 0, ’ senora ’: 0, ’ lass ’: 0, ’ beauty ’: 0, ’
+ingenue ’: 0, ’ belle ’: 0, ’ doll ’: 0, ’ se˜nora ’: 0, ’ senorita
+’: 0, ’ lassie ’: 0, ’ ing´enue ’: 0, ’ miss ’: 0, ’ mademoiselle
+’: 0, ’ se˜norita ’: 0, ’ babe ’: 0, ’ girlfriend ’: 0, ’ lover ’: 0, ’
+mistress ’: 0, ’ ladylove ’: 0, ’ inamorata ’: 0, ’ gill ’: 0, ’ old
+’: 2, ’ beloved ’: 0, ’ dear ’: 0, ’ sweetheart ’: 0, ’ sweet ’: 0,
+’ flame ’: 2, ’ love ’: 5, ’ valentine ’: 0, ’ favorite ’: 1, ’ moll
+’: 0, ’ darling ’: 0, ’ honey ’: 0, ’ significant ’: 0, ’ wife ’: 3,
+’ wifey ’: 0, ’ missus ’: 0, ’ helpmate ’: 0, ’ helpmeet ’: 0, ’
+spouse ’: 0, ’ bride ’: 1, ’ partner ’: 0, ’ missis ’: 0, ’ widow
+’: 0, ’ housewife ’: 0, ’ mrs ’: 0, ’ matron ’: 0, ’ soul ’: 3, ’
+mate ’: 1, ’ housekeeper ’: 0, ’ dowager ’: 0, ’ companion ’: 0,
+’ homemaker ’: 0, ’ consort ’: 0, ’ better half ’: 0, ’ hausfrau
+’: 0, ’ stay-at-home ’: 0, ’ he ’: 80, ’ him ’: 49, ’ boy ’: 3,
+’ man ’: 1, ’ male ’: 10, ’ guy ’: 1, ’ masculine ’: 0, ’ virile
+’: 0, ’ manly ’: 0, ’ man-sized ’: 0, ’ hypermasculine ’: 0, ’
+macho ’: 0, ’ mannish ’: 0, ’ manlike ’: 0, ’ man-size ’: 0, ’
+hairy-chested ’: 0, ’ butch ’: 0, ’ ultramasculine ’: 0, ’ boyish
+’: 0, ’ tomboyish ’: 0, ’ hoydenish ’: 0, ’ amazonian ’: 0, ’
+gentleman ’: 0, ’ dude ’: 0, ’ fellow ’: 0, ’ cat ’: 2, ’ gent ’:
+0, ’ fella ’: 0, ’ lad ’: 0, ’ bloke ’: 0, ’ bastard ’: 0, ’ joe ’: 0,
+’ chap ’: 0, ’ chappie ’: 0, ’ hombre ’: 0, ’ galoot ’: 0, ’ buck
+’: 0, ’ joker ’: 3, ’ mister ’: 0, ’ jack ’: 8, ’ sir ’: 0, ’ master
+’: 1, ’ buddy ’: 0, ’ buster ’: 0], ’racial’:... }
+
+Fig. 13. WandB parallel coordinates for Swedish BERT training on MAB-Swedish.
+
+learning_rate
+f1
+num_train_epochs
+0.00090
+0°6
+0.86930
+8.8
+0.86920
+0.00080
+8.6
+0.86910
+8.4 -
+0.00070
+0.86900
+8.2
+0.86890
+0.00060
+8.0-
+0.86880
+7.8
+0.00050
+Q.86870
+7.6
+7.4
+0.86860
+0.00040
+7.2
+0.86850
+0.00030
+7.0
+0.86840
+6.8
+0.86830
+0.00020
+6.6
+0.86820
+6.4 
+0.00010
+0.86810
+6.2
+0.00000
+6.0
+0.86800

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+arXiv:2301.01691v1  [physics.plasm-ph]  4 Jan 2023
+Special behavior of alkali beam emission spectroscopy in low-ion-temperature plasma
+P. Bal´azs1,2, O. Asztalos1,2, G. Anda2, M. Vecsei2, S. Zoletnik2, S. T. A. Kumar3, G. I. Pokol1,2
+1Institute of Nuclear Techniques, Budapest University of Technology and Economics, Budapest, Hungary
+2Centre for Energy Research, Budapest, Hungary and
+3Department of Electrical and Computer Engineering, HSX Plasma Laboratory,
+University of Wisconsin-Madison, Madison, WI, United States of America
+Beam emission spectroscopy (BES) is a powerful plasma diagnostic method especially suited for
+the measurement of plasma density and its fluctuations. Designing a BES system for an experimental
+fusion device is, however, not trivial, and the process can be greatly aided by computer simulations
+of how a proposed setup would work in the given environment. This paper presents such analysis
+utilizing the RENATE-OD synthetic diagnostic code for a proposed alkali BES system on the HSX
+stellarator. HSX is a device featuring an unusual operating regime in the world of fusion devices
+due to the low ion temperature and low plasma density. It was found that BES shows unusual
+tendencies in these conditions. The relation between beam energy and plasma penetration in low-
+ion-temperature plasma, together with unique emission features facilitated by low-density plasma,
+and the underlying reasons behind these features are explored in this paper.
+I.
+INTRODUCTION
+In the current development stage of fusion technology,
+plasma diagnostic tools are still much needed to sup-
+port the validation of various theories and verification
+of engineering solutions.
+Active diagnostic techniques,
+when some kind of probing of the plasma is carried out,
+are especially attractive due to their localized measure-
+ments.
+Such a system is beam emission spectroscopy
+(BES), used in several experimental devices, including
+JET [1], ASDEX Upgrade [2], EAST [3], DIII-D [4],
+and even small devices like COMPASS [5]. This tech-
+nique probes the plasma with a neutral atomic beam,
+resulting in the release of light due to interactions with
+the energetic plasma particles. The wavelength of this
+light is characteristic to the beam material, which can be
+one of the hydrogen isotopes in larger beams, or alkali
+metals (lithium, sodium) in the small diagnostic beams.
+The intensity of the emitted light is highly dependent on
+the plasma density, which combined with the ability of
+continuous operation makes the technique ideal for spa-
+tially and temporally resolved density fluctuation mea-
+surements. This is especially true for alkali diagnostic
+beams, with the added benefit of the emission spectrum
+being well distinguishable from the background. The lim-
+itation of this type of BES is the generally short, few cen-
+timeters long penetration into the plasma, but the region
+spanned by this limitation is still interesting in terms of
+transport and magnetohydrodynamic phenomena.
+The construction of a BES system requires careful de-
+sign due to geometric restrictions and the desire to max-
+imize the collectable photon flux [6]. As an aid for this
+problem, one can utilize BES simulation codes designed
+to produce synthetic measurements under the specific
+plasma conditions of the device in question. Based on
+the data of such simulations it is possible to analyze
+the expected performance of the diagnostic system, help-
+ing the engineers to balance between design trade-offs.
+RENATE-OD [7] (Rate Equations for Neutral Alkali-
+beam TEchnique - Open Development) is an open-source
+Python package developed for this purpose. This code
+solves rate equations to calculate the population of each
+atomic level considered in the calculations, from which
+the emissivity of the beam is acquired through the spon-
+taneous transition rate for the observed spectral line.
+With the ability to accurately predict the performance
+of a BES system under design, it is possible to dynami-
+cally test various injection and observation configurations
+for optimal operation.
+RENATE-OD and its precursor, RENATE [8], has al-
+ready been utilized in multiple projects regarding BES
+systems. The code’s capabilities have been validated with
+KSTAR measurements [9, 10], it was the main tool for
+a feasibility study of a BES system for JT60-SA [11],
+and aided the design of lithium beam diagnostics on
+EAST [12]. Recently, we performed a feasibility study for
+the HSX (Helically Symmetric eXperiment) stellarator.
+This device is operated by the Electrical and Computer
+Engineering department of the University of Wisconsin-
+Madison, with the aim of investigating transport, tur-
+bulence, and confinement in a quasi-helically symmetric
+magnetic field [13]. It is a small device with average ma-
+jor and minor radii of 1.2 m and 0.15 m, respectively. The
+magnetic field is produced by copper coils, allowing dis-
+charges up to only 100 ms long, and a maximum on-axis
+magnetic field of 1.25 T. The plasma density achievable in
+the device is on the order of a few times 1018 m−3, which
+is relatively low compared to other fusion-related plasma
+experiments. As a consequence, the electrons heated by
+electron cyclotron resonance are poorly coupled to the
+ions, resulting in an electron and ion population with dis-
+parate temperatures. Throughout this paper, we treated
+the ion temperature as constant at 50 eV throughout the
+machine, while the electron temperature was prescribed
+with a peaked profile reaching a maximum of 2.5 keV
+[14].
+As of now, the device does not have a BES system,
+therefore we performed a feasibility study to explore how
+the technique would perform under such conditions. Part
+of this process was to simulate the evolution of lithium
+
+2
+(Li) and sodium (Na) beams across low-density and low-
+ion-temperature plasma profiles as described before. We
+observed that both the beam density and emissivity can
+behave unusually under these circumstances.
+Notably,
+a beam with low energy could be less attenuated than
+a high-energy beam, and changes in the electron tem-
+perature are more visible in the emission compared to
+measurements in high-density plasma. The reasons be-
+hind these phenomena are explored in this work.
+We
+also note that these plasma conditions are not exclusive
+to HSX, since they can be regularly found in the diver-
+tor region of larger fusion devices. Measurement of these
+regions by BES is an unexplored area, which warrants
+further efforts for investigation.
+The basic methodology of the calculations performed
+by RENATE-OD are presented in Chapter II, then in
+Chapter III the unusual effects found in beam attenua-
+tion and emission are presented together with their ex-
+planation. Finally, in Chapter IV the findings and their
+relevance are summarized.
+II.
+CALCULATIONS
+To simulate the density and emission of an atomic
+beam along its path, RENATE-OD solves the rate equa-
+tions describing the population of the valence electrons
+of the beam atoms on the most populated atomic levels.
+In the case of alkali atoms, the considered levels are the
+l-resolved states reachable by the valence electron up to
+4.5 eV in the case of lithium, which includes 2s, 2p, 3s,
+3p, 3d, 4s, 4p, 4d, 4f, and up to 4.1 eV for sodium, in-
+cluding 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s. The populations
+of the levels evolve due to interactions with the plasma
+components. The simulated light emission is governed by
+collisional excitation to the higher energy levels followed
+by spontaneous emission. The attenuation of the beam
+is contributed to the beam atoms becoming charged, and
+therefore redirected by the perpendicular magnetic field
+component. This can happen through collisional ioniza-
+tion or charge exchange with plasma ions, and we refer
+to the sum of these processes as electron loss. The simu-
+lation does not include interaction between beam atoms
+due to being negligible compared to beam-plasma inter-
+actions. It is also assumed that only the valence electrons
+participate in the relevant processes, and initially, all of
+them are in the ground state.
+The atomic levels can gain or lose electrons through
+multiple channels. For example, if we denote the popu-
+lation density of a particular level with ni, we can write
+its losses due to excitation by electrons in the form of
+�dni
+dt
+�
+el,exc
+= −nine
+�
+i<j
+Re,exc
+i→j (T ) ,
+(1)
+where ne is the electron density in the plasma,
+Re,exc
+i→j (T ) is the rate coefficient expressing the probability
+of electron excitation from level i to j, and T is plasma
+temperature. The temperature dependence of the rate
+coefficient originates from the way it is calculated with
+the integral
+R = ⟨σv⟩ =
+�
+σ(v)vf(v, T )dv ,
+(2)
+where R is a general rate coefficient, σ(v) is the cross-
+section of the reaction, v is the relative velocity of the
+beam atom and plasma particle, and f(v, T ) is the ve-
+locity distribution of the plasma particles.
+The cross-
+sections for the different reactions are based on [15–17]
+for lithium, and on [18] for sodium.
+If we collect all the terms similar to the electron ex-
+citation term in equation (1), namely excitation, de-
+excitation, and electron loss caused by all plasma com-
+ponents, we get an equation for the temporal evolution
+of ni. However, we would like to express the properties
+of the beam spatially along its path (x). Assuming that
+the beam atoms travel with a constant speed (vb) the
+conversion is simply
+dni
+dx = 1
+vb
+dni
+dt .
+(3)
+In this sense, the spatial evolution of a beam can be
+expressed with the reduced rate coefficients, which are
+the regular rate coefficients divided by the beam velocity:
+¯R = R
+vb
+.
+(4)
+This is the main quantity we study closely when we
+want to explain specific phenomena in beam evolution.
+Its values were precalculated for each type of interaction
+for plasma temperatures in the range of 1 − 105 eV, so
+during the simulation of beams ¯R is looked up according
+to the local plasma temperature for each point of the
+beam.
+III.
+RESULTS
+In order to perform the necessary beam simulations for
+the HSX study, the plasma composition together with the
+density and temperature profiles were required as inputs.
+We assumed a pure hydrogen plasma as a simple test
+case, which also meant equal proton and electron density
+in accordance with charge neutrality. The profiles were
+based on previous publications[14] on the results of the
+experiment. Figure 1 shows the radial electron density
+and temperature profiles, with the minor radius (r) nor-
+malized to the device’s minor radius (a). For ions, the
+same density profile was used, and the ion temperature
+was assumed to be homogeneously 50 eV according to
+[19]. This is different from the conditions found in other
+
+3
+0.0
+0.2
+0.4
+0.6
+0.8
+r/a
+0
+1
+2
+3
+4
+Electron density [m
+−3
+]
+1e18
+a)
+0.0
+0.2
+0.4
+0.6
+0.8
+r/a
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Electron temperature [keV]
+b)
+FIG. 1. Electron density (a) and temperature (b) profiles based on [14].
+0.1
+0.2
+0.3
+Distance alo g beam [m]
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Normalized beam de sity
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Electro  de sity [m
+−3
+]
+1e18
+Li beam de sity
+10
+20
+30
+40
+50
+60
+Beam e ergy [keV]
+a)
+0.1
+0.2
+0.3
+Distance al ng beam [m]
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+N rmalized beam density
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Electr n density [m
+−3
+]
+1e18
+Na beam density
+10
+20
+30
+40
+50
+60
+Beam energy [keV]
+b)
+FIG. 2. The density of lithium (a) and sodium (b) beams in HSX-like plasma normalized to the initial beam density (solid
+lines). Beam energy is represented by the coloring of the lines. The electron density of the plasma is also shown with the red,
+dashed line. Signs of the inverse relation between beam energy and penetration depth are observable in both cases.
+devices because usually, a higher plasma density ensures
+that the electron and ion population has a similar tem-
+perature.
+A.
+Beam density
+The evolution of lithium and sodium beams calculated
+by RENATE-OD for various energies are plotted in Fig-
+ure 2. Normalized beam density values are used, which
+are acquired through normalization by the initial beam
+density. The beam energies applied are in the range of
+10-60 keV, which is the usual operating range of alkali
+diagnostic beams.
+The first detail worth noting is the overall attenua-
+tion of the beams across the whole plasma. The beam
+density at the far end of the simulated region is still 50%-
+62% of the initial density, meaning that a significant por-
+tion of the beam atoms would reach the opposite wall
+from the injection site. Also, in a more usual high-ion-
+temperature plasma, we expect the penetration depth to
+increase with beam energy [20], which is still the situation
+for lithium beams above 25 keV acceleration. However,
+lithium beams under 25 keV and sodium beams in the
+whole energy range exhibit an inverse relation, meaning
+the beams with lower energy can penetrate the plasma
+better.
+In Figure 3 the last normalized beam density values
+of the simulated region are plotted against beam energy,
+showing the remaining portion of the beam atoms after
+traveling through the plasma.
+In the case of lithium,
+the usual tendency of less attenuation with increasing
+beam energy still holds between 25 and 60 keV, but the
+situation reverses under 25 keV. For sodium beams, the
+whole simulated energy range shows the inverse tendency.
+This behavior can be better understood by examin-
+
+4
+10
+20
+30
+40
+50
+60
+Beam energy [keV]
+0.50
+0.52
+0.54
+0.56
+0.58
+0.60
+0.62
+Normalized beam density
+Last value of the
+normalized beam density for Li
+a)
+10
+20
+30
+40
+50
+60
+Beam energy [keV]
+0.48
+0.50
+0.53
+0.55
+0.57
+0.60
+0.62
+Normalized beam density
+Last value of the
+normalized beam density for Na
+b)
+FIG. 3. Final values of the normalized beam density at the end of the simulated region. Increasing penetration depth with
+decreasing energy is only present under 25 keV beam energy with lithium (a), while with sodium this is the case for the whole
+energy range (b).
+10
+1
+10
+3
+10
+5
+Proton tem erature [eV]
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Reduced rate coefficient [cm
+2
+]
+1e−14
+Proton im act electron loss from Na(3s)
+10
+20
+30
+40
+50
+60
+Beam energy [keV]
+FIG. 4.
+Reduced rate coefficients calculated with equations
+(2) and (4) for proton impact electron loss from Na(3s) (solid
+lines). The different beam energies are represented by differ-
+ent colors. The dashed lines mark 50 eV and 1 keV proton
+temperature.
+ing the underlying cross-sections and the corresponding
+reduced rate coefficients. In particular, it is enough to
+study the electron loss process from the ground state of
+the beam atoms, since this is the main process behind
+beam attenuation, and processes of the higher levels usu-
+ally follow similar trends as those of the ground level.
+Additionally, the large difference between the tempera-
+ture of the ion and electron population in the plasma
+causes the attenuation to be dominated by proton im-
+pacts, therefore the analysis is concentrated on this pro-
+cess. The following section describes the corresponding
+rate coefficients for sodium beams, but the main effects
+are similar in nature for the lithium beams, too.
+The reduced rate coefficients are shown in Figure 4 as
+continuous functions of ion temperature for the differ-
+ent beam energies. At the ion temperature of the HSX
+plasma, marked with the red and dashed line, beam at-
+tenuation increases with beam energy, which explains the
+results seen in Figure 3. It is also visible that increasing
+the ion temperature, for example up to 1 keV marked by
+the green and dashed line, causes this relation to reverse.
+To understand the connection between the reduced
+rate coefficients and ion temperature, we have to exam-
+ine the quantities found in the integral of equation (2).
+The σ(v)v product and separately the distribution func-
+tion f(v, T ) for the proton impact electron loss rate from
+the 3s level of 10 keV sodium atoms are shown in Fig-
+ure 5/a. For the illustration of how plasma temperature
+changes the rate coefficients, there are two distribution
+functions are plotted, one according to 50 eV and one
+according to 1 keV ion temperature. Note that both of
+these functions are centered on the same relative veloc-
+ity determined by the beam energy. When the plasma
+temperature is low, the distribution function resembles a
+Dirac delta, so the rate coefficient is relatively low. How-
+ever, increasing the temperature widens the distribution
+function, which combined with the convex nature of the
+σ(v)v product increases the rate coefficient.
+On the other hand, the situation is different when we
+perform these calculations for a beam with higher, 60
+keV energy. The same curves are plotted in Figure 5/b
+for this case. It is immediately visible that the distri-
+bution functions are shifted to a higher velocity. This
+has two effects. First, with 50 eV plasma temperature,
+the resulting rate coefficient in Figure 4 is higher for the
+high energy beam, since the Dirac-delta-like distribution
+function selects a higher value from the σ(v)v product.
+However, in this velocity range the curve of this prod-
+
+5
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+Velocity [m/s]
+1e6
+0
+1
+2
+3
+4
+5
+σ(v)v [cm
+2
+m/s]
+1e−9
+σ(v)v
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Veloci y dis ribu ion [a.u.]
+f(v) a  10 keV
+50 eV
+1000 eV
+a)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+Velocity [m/s]
+1e6
+0
+1
+2
+3
+4
+5
+σ(v)v [cm
+2
+m/s]
+1e−9
+σ(v)v
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Veloci y dis ribu ion [a.u.]
+f(v) a  60 keV
+50 eV
+1000 eV
+b)
+FIG. 5. a) Cross-section of electron loss from the 3s level of sodium due to proton impacts, the main channel behind beam
+attenuation, multiplied with the relative velocity as in equation (2). The highlighted section is the velocity range covered by the
+simulated beam energies. The remaining term under the integral in formula (2), the relative velocity distribution function, is
+also plotted for both 50 eV and 1 keV plasma temperature for a 10 keV plasma temperature beam. b) The same curves are also
+plotted for a sodium beam with 60 keV energy.
+uct is concave, so using the wider distribution function
+of 1 keV plasma protons only slightly increases the rate
+coefficient.
+The temperature dependence of proton impact electron
+loss rate coefficients and the low ion temperature explain
+the inverse relation between beam energy and penetra-
+tion depth in the HSX study.
+Usually, neutral beams
+operate in plasma where beam attenuation is driven by
+both a high-temperature ion and high-temperature elec-
+tron population, leading to low-energy beams suffering
+from higher electron loss rates.
+However, in the low-
+ion-temperature and high-electron-temperature plasma
+of HSX, this tendency reverses completely for sodium
+beams, and partially for lithium beams in the examined
+energy region, meaning that low-energy beams experi-
+ence lower electron loss rates.
+B.
+Beam emission
+Understanding the evolution of beam density is an im-
+portant step when evaluating the simulation results, but
+the quantity more directly related to the performance of
+the BES diagnostic system is the emission density of the
+beam. This is acquired by multiplying the population
+density of the upper level of the transition we are in-
+terested in with the spontaneous transition rate to the
+lower level. Since we calculated one-dimensional beam
+evolution, this gives us the linear emission density of the
+simulated beam.
+This quantity is plotted for the previously discussed
+sodium beams in Figure 6. The transition we considered
+is 3p → 3s, as the one observed in existing diagnostic sys-
+tems [21]. As expected, beams with lower energy provide
+higher peak emission densities due to their lower veloc-
+0.1
+0.2
+0.3
+Distance along beam [m]
+0
+1
+2
+3
+Linea  emission density [s
+−1
+m
+−1
+]
+1e16
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+No malized p ofiles
+Na linea  emission density
+T
+e
+n
+e
+10
+20
+30
+40
+50
+60
+Beam ene gy [keV]
+FIG. 6. Linear emission density of sodium beams with differ-
+ent energy (solid lines). The electron temperature and den-
+sity profiles are also plotted with red and black dashed lines,
+respectively. Beams with lower energy emit more light, which
+is expected due to the lower speed of the beam atoms, giving
+more chance for excitation over a given distance. However,
+such a strong reaction to the increase in electron temperature
+is rarely seen, since usually beam evolution is mostly driven
+by plasma density.
+ities through the plasma, providing a higher probability
+for excitation over a given distance.
+The evolution of the emission density is fairly unusual,
+with high emission across the whole plasma, and peaks
+on both sides of the core. The high emission is evidently
+due to the low plasma density, and therefore low beam at-
+tenuation, as seen in the previous section. Regarding the
+
+6
+10
+1
+10
+3
+10
+5
+Electron tempe atu e [eV]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Reduced  ate coefficient [cm
+2
+]
+1e−14
+Elect on impact excitation f om Na(3s) to Na(3p)
+10
+20
+30
+40
+50
+60
+Beam ene gy [keV]
+FIG. 7.
+Reduced rate coefficients in the function of elec-
+tron temperature for electron impact induced excitation from
+Na(3s) to Na(3p), the main reaction channel populating the
+3p level. In the temperature range of ∼100-2500 eV present
+in the HSX plasma, the rate coefficients are decreasing for all
+beam energies.
+two distinct emission peaks, the explanation lies within
+the temperature dependence of rate coefficients govern-
+ing the population of the 3p level. In particular, the most
+dominant reaction populating this level is electron impact
+excitation from the 3s level. The reduced rate coefficients
+for this process are plotted in Figure 7 for different beam
+energies against electron temperature. In the tempera-
+ture region of the HSX plasma profile (∼100-2500 eV),
+the rate coefficients are steadily decreasing with increas-
+ing temperature, which explains the behavior seen in
+the emission profile. As the beam is traveling through
+the plasma, first the emission is increasing as the higher
+atomic levels get populated due to the increasing plasma
+density. However, as the beam reaches the core plasma,
+there is a rapid increase in electron temperature, lead-
+ing to a significant decrease in the rate at which the 3p
+level gets populated. This is mirrored by a decrease in
+the emission as well.
+After traveling through the core
+plasma, the electron temperature drops again, leading to
+an increase in the beam emission.
+Of course, this effect is always present, but the plasma
+in HSX is unique due to its low density and low beam
+attenuation, allowing the emission to maintain a high
+intensity throughout the plasma, and also due to its ex-
+tremely peaked electron temperature profile, making a
+strong impact on the emission profile.
+IV.
+SUMMARY AND OUTLOOK
+Recently, we completed a feasibility study for the HSX
+stellarator, which has the unique trait among fusion ex-
+periments of studying low-ion-temperature plasma con-
+figurations.
+In the study, we examined the expected
+performance of an alkali beam emission diagnostic sys-
+tem by simulating the beam evolution with our in-house
+code, RENATE-OD. We performed simulations for both
+lithium and sodium beams in the energy range of 10-60
+keV, propagating through a plasma profile acquired from
+earlier experiments.
+The results, namely the relation between plasma pen-
+etration and beam energy, showed unusual effects com-
+pared to other BES diagnostics working in high-ion-
+temperature plasma. Instead of the penetration increas-
+ing with beam energy in the whole energy region, we
+found that low-energy beams achieve better penetration.
+The explanation for this effect is in the underlying elec-
+tron loss cross-sections and the corresponding rate coef-
+ficients.
+The temperature-dependent tendencies of the
+rate coefficients are determined by the curvature of the
+σ(v)v product in the relevant relative velocity region dur-
+ing the integration with the ion distribution function.
+The center of the relevant region is at the velocity of
+beam atoms, and its width is determined by the temper-
+ature of the ion population, so both beam energy and
+ion temperature may have a significant effect on the rate
+coefficients. In case of the HSX-like conditions, this re-
+lation manifests in low-energy beams experiencing lower
+attenuation, than the high-energy beams of the study.
+Apart from beam attenuation, we also examined the
+beam emission density, as the most important quantity
+for the capabilities of the BES diagnostic system. The
+unusual plasma parameters manifested some interesting
+effects in this case as well. First, the low plasma density
+allows the emission density to remain high throughout
+the plasma, unlike in high-density plasma configurations,
+where the emission reaches its peak and decays in a few
+centimeters. This also allows the peaked electron tem-
+perature profile to make its impact on the beam emis-
+sion, since the high temperature in the core lowers the
+rate of the exited states getting populated, resulting in
+decreased emission density in the core.
+Altogether, it is clear that the diagnostics of low-ion-
+temperature and low-density plasma with beam emission
+spectroscopy feature effects unseen in high-density exper-
+iments. While the former conditions are far from those
+of a fusion-supporting plasma, the outer regions and the
+divertor vicinity of future large-scale devices, equipped
+with more exotic divertor configurations like the Super-
+X in MAST-U, are expected to hold plasma resembling
+the HSX conditions [22–24]. These regions form the in-
+terface between the plasma and the chamber, so monitor-
+ing them is important for both scientific and operational
+purposes. If BES is ever used for the diagnostics of these
+areas, the effects described in our study should be con-
+sidered carefully.
+
+7
+ACKNOWLEDGMENTS
+Supported by the KDP-2021 Program of the Ministry
+for Innovation and Technology from the source of the
+National Research, Development and Innovation Fund.
+G.I. Pokol, P. Bal´azs and O. Asztalos acknowledge the
+support of the National Research, Development and In-
+novation Office (NKFIH) Grant FK132134.
+This work has been carried out within the framework
+of the EUROfusion Consortium, funded by the Euro-
+pean Union via the Euratom Research and Training Pro-
+gramme (Grant Agreement No 101052200 - EUROfu-
+sion). Views and opinions expressed are however those of
+the author(s) only and do not necessarily reflect those of
+the European Union or the European Commission. Nei-
+ther the European Union nor the European Commission
+can be held responsible for them.
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+

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+arXiv:2301.05288v1  [cs.MA]  12 Jan 2023
+Springer Nature 2021 LATEX template
+An Approach to Stochastic Dynamic Games
+with Asymmetric Information and Hidden
+Actions
+Yi Ouyang1*, Hamidreza Tavafoghi2 and Demosthenis
+Teneketzis3
+1Preferred Networks America, Inc., Burlingame, CA, USA.
+2Google, Mountain View, CA, USA.
+3Department of Electrical Engineering and Computer Science,
+University of Michigan, Ann Arbor, MI, USA.
+*Corresponding author(s). E-mail(s): ouyangyii@gmail.com;
+Contributing authors: hamidreza.tavafoghi@gmail.com;
+teneket@umich.edu;
+Abstract
+We consider in discrete time, a general class of sequential stochastic
+dynamic games with asymmetric information with the following features.
+The underlying system has Markovian dynamics controlled by the agents’
+joint actions. Each agent’s instantaneous utility depends on the current
+system state and the agents’ joint actions. At each time instant each
+agent makes a private noisy observation of the current system state and
+the agents’ actions in the previous time instant. In addition, at each time
+instant all agents have a common noisy observation of the current sys-
+tem state and their actions in the previous time instant. Each agent’s
+actions are part of his private information. The objective is to determine
+Bayesian Nash Equilibrium (BNE) strategy profiles that are based on a
+compressed version of the agents’ information and can be sequentially
+computed; such BNE strategy profiles may not always exist. We present
+an approach/methodology that achieves the above-stated objective,
+along with an instance of a game where BNE strategy profiles with the
+above-mentioned characteristics exist. We show that the methodology
+also works for the case where the agents have no common observations.
+Keywords: Dynamic games, asymmetric information, hidden actions,
+common information, information compression, sequential decomposition
+1
+
+Springer Nature 2021 LATEX template
+2
+Dynamic Games with Asymmetric Information and Hidden Actions
+1 Introduction
+We study, in discrete time, a general class of sequential stochastic dynamic
+games with asymmetric information. We consider a setting where the under-
+lying system has Markovian dynamics controlled by the agents’ joint actions.
+Each agent’s instantaneous utility depends on the agents’ joint actions and the
+system state. At each time instant each agent makes a private noisy observa-
+tion that depends on the current system state and the agents’ actions in the
+previous time instant. In addition, at each time instant all agents may have a
+common noisy observation of the system state and their actions in the previous
+time instant. The agents’ actions are hidden, that is, each agent’s actions are
+not directly observable by the other agents. Therefore, at every time instant
+agents have asymmetric and imperfect information about the game’s history.
+Dynamic games with the above features arise in engineering (cybersecurity,
+transportation, energy markets), in economics (industrial organization), and
+in socio-technological applications.
+As pointed out in Tang et al (2022), the key challenges in the study of
+dynamic games with asymmetric information are: (i) The domain of agents’
+strategies increases with time, as the agents acquire information over time.
+Thus, the computational complexity of the agents’ strategies increases with
+time. (ii) Due to signaling1 (Ho, 1980), in many instances an agent’s assess-
+ment of the game’s status at time t, therefore his strategy at time t, depends
+on the strategies of agents who acted before him. Consequently, we cannot
+obtain the standard sequential decomposition (that sequentially determines
+the components of an equilibrium strategy profile) of the kind provided by the
+standard dynamic programming algorithm (where the agent’s optimal strat-
+egy at any time t does not depend on past strategies (Kumar and Varaiya,
+1986, Chapter 6.5)).
+To address these challenges, we can look for equilibrium strategy profiles
+that are based on a compressed version of the agents’ information and can be
+sequentially computed. However, such equilibrium strategy profiles may not
+exist.
+In this paper we propose an approach, described in detail in Section 3, that
+addresses the above-stated challenges. According to this approach, we first
+compress the agents’ private and common information at each time instant.
+Then, we define strategies based on the compressed information and show that
+Bayesian Nash Equilibria (BNE) based on these strategies can be determined
+sequentially in time moving backwards, if each step of this backwards proce-
+dure has a solution. Finally, we provide an example where a BNE strategy
+profile based on compressed information exists.
+We show that the proposed approach works for the case where the agents
+have no common observations and their actions are hidden.
+1Signaling in games is more complex than signaling in teams because the agents have diverging
+incentives and their strategies are their own private information.
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+3
+1.1 Related Literature
+Dynamic games with asymmetric information have been extensively investi-
+gated in the literature in the context of repeated discounted games; see Zamir
+(1992); Forges (1992); Aumann et al (1995); Mailath and Samuelson (2006)
+and the references therein. The key feature of these games is the absence of a
+dynamic system. Moreover, the works on repeated games study primarily their
+asymptotic properties when the horizon is infinite and agents are sufficiently
+patient (i.e. the discount factor is close one). In repeated games, agents play
+a stage (static) game repeatedly over time. The main objective of this strand
+of literature is to explore situations where agents can form self-enforcing pun-
+ishment/reward mechanisms so as to create additional equilibria that improve
+upon the payoffs they can get by simply playing an equilibrium of the stage
+game over time. Recent works (see H¨orner et al (2011); Escobar and Toikka
+(2013); Sugaya (2012)) adopt approaches similar to those used in repeated
+games to study infinite horizon dynamic games with asymmetric information
+when there is an underlying dynamic Markovian system. Under certain condi-
+tions on the system dynamics and information structure, the authors of H¨orner
+et al (2011); Escobar and Toikka (2013); Sugaya (2012) characterize a set of
+asymptotic equilibria attained when the agents are sufficiently patient.
+The problem we study in this paper is different from the ones in Zamir
+(1992); Forges (1992); Aumann et al (1995); Mailath and Samuelson (2006);
+H¨orner et al (2011); Escobar and Toikka (2013); Sugaya (2012) in two aspects.
+First, we consider a class of dynamic games where the underlying system
+has general Markovian dynamics and a general information structure, and we
+do not restrict attention to asymptotic behaviors when the horizon is infi-
+nite and the agents are sufficiently patient. Second, we study situations where
+the decision problem that each agent faces, in the absence of strategic inter-
+actions with other agents, is a Partially Observed Markov Decision Process
+(POMDP), which is a complex problem to solve by itself. Therefore, reach-
+ing (and computing) a set of equilibrium strategies, which take into account
+the strategic interactions among the agents, is a very challenging task. As a
+result, it is not very plausible for the agents to seek reaching equilibria that
+are generated by the formation of self-enforcing punishment/reward mecha-
+nisms similar to those used in infinitely repeated games. We believe that our
+results provide new insight into the behavior of strategic agents in complex and
+dynamic environments, and complement the existing results in the repeated
+games literature.
+Stochastic dynamic zero-sum games with asymmetric information have
+been studied in Renault (2006); Cardaliaguet et al (2015); Gensbittel and
+Renault (2015); Li et al (2017); Kartik and Nayyar (2021); Zheng and Casta˜n´on
+(2013); Li and Shamma (2014). The authors of Renault (2006); Cardaliaguet
+et al (2015); Zheng and Casta˜n´on (2013); Li and Shamma (2014) study zero-
+sum games with Markovian dynamics and lack of information on one side
+(i.e. one informed and one uninformed agent). The authors of Gensbittel and
+Renault (2015); Li et al (2017); Kartik and Nayyar (2021) study zero-sum
+
+Springer Nature 2021 LATEX template
+4
+Dynamic Games with Asymmetric Information and Hidden Actions
+games with Markovian dynamics and lack of information on both sides. The
+works of Renault (2006); Cardaliaguet et al (2015); Gensbittel and Renault
+(2015); Li et al (2017); Kartik and Nayyar (2021); Zheng and Casta˜n´on
+(2013); Li and Shamma (2014) consider specific information structures. Specif-
+ically: the actions of both agents are publicly observed; in Renault (2006);
+Cardaliaguet et al (2015); Zheng and Casta˜n´on (2013); Li and Shamma (2014)
+the informed agent observes perfectly the state of the dynamic system, the
+other agent has no direct observation of the system’s state; in Gensbittel
+and Renault (2015); Li et al (2017) each agent observes perfectly part of the
+system’s state and the states observed by the two agents are either indepen-
+dent or conditionally independent (given the observed actions). The authors
+of Kartik and Nayyar (2021) consider a general information structure where
+each agent has some private information and the agents share some infor-
+mation about the dynamic system’s state and their actions. The authors of
+Renault (2006); Cardaliaguet et al (2015); Gensbittel and Renault (2015); Li
+et al (2017); Kartik and Nayyar (2021); Zheng and Casta˜n´on (2013); Li and
+Shamma (2014) derive their results by taking advantage of properties of zero-
+sum games such as the interchangeability of equilibrium strategies and the
+unique value of the game. These properties do not extend to non-zero sum
+games. We study a general class of stochastic dynamic games that include
+zero-sum stochastic dynamic games with asymmetric information as a special
+case. We consider general Markovian dynamics for the underlying system in
+contrast to Renault (2006); Cardaliaguet et al (2015); Gensbittel and Renault
+(2015); Li et al (2017); Zheng and Casta˜n´on (2013); Li and Shamma (2014),
+where the system has the special structure described above. We consider a
+general information structure that allows us to capture scenarios with unob-
+servable actions and imperfect observations that are not captured by Renault
+(2006); Cardaliaguet et al (2015); Gensbittel and Renault (2015); Li et al
+(2017); Zheng and Casta˜n´on (2013); Li and Shamma (2014).
+The problems investigated in Tang et al (2022); Nayyar et al (2014); Gupta
+et al (2014); Ouyang et al (2015, 2017); Vasal and Anastasopoulos (2016);
+Sinha and Anastasopoulos (2016); Gupta et al (2016); Nayyar et al (2013a)
+are the most closely related to our problem. The authors of Nayyar et al
+(2014); Gupta et al (2014, 2016); Nayyar et al (2013a) study a class of dynamic
+games where the agents’ common information based belief (defined in Nayyar
+et al (2014)) is independent of their strategies, that is, there is no signaling
+among them. This property allows them to apply ideas from the common
+information approach developed in Nayyar et al (2011, 2013b), and define an
+equivalent dynamic game with symmetric information among fictitious agents.
+Consequently, they characterize a class of equilibria for dynamic games called
+Common Information based Markov Perfect Equilibria.
+Our results are different from those in Nayyar et al (2014); Gupta et al
+(2014, 2016); Nayyar et al (2013a) in two aspects. First, we consider a general
+class of dynamic games where the agents’ CIB beliefs are strategy-dependent,
+thus, signaling is present. Second, the proposed approach in Nayyar et al
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+5
+(2014); Gupta et al (2014, 2016); Nayyar et al (2013a) requires the agents
+to keep track of all of their private information over time. We propose an
+approach to effectively compress the agents’ private information, and conse-
+quently, reduce the number of variables which the agents need to form CIB
+beliefs.
+The authors of Tang et al (2022); Ouyang et al (2015, 2017); Vasal and
+Anastasopoulos (2016); Sinha and Anastasopoulos (2016) study a class of
+dynamic games with asymmetric information where signaling occurs. When the
+horizon in finite, the authors of Ouyang et al (2015, 2017) introduce the notion
+of Common Information Based Perfect Bayesian Equilibrium, and provide a
+sequential decomposition of the game over time. The authors of Vasal and
+Anastasopoulos (2016); Sinha and Anastasopoulos (2016) extend the results of
+Ouyang et al (2015, 2017) to finite horizon Linear-Quadratic-Gaussian (LQG)
+dynamic games and infinite horizon dynamic games, respectively.
+The work of Tang et al (2022) extends the model of Ouyang et al (2017)
+to games among teams of agents. Each agent has his own private informa-
+tion which he shares with the members of his own team with delay d; teams
+also have common information. The authors of Tang et al (2022) consider two
+classes of strategies: sufficient private information based (SPIB) strategies,
+which only compress private information, and sufficient private and common
+information based (SPCIB) strategies, which compress both common and pri-
+vate information. They show that SPIB-strategy-based BNE exist and the set
+of payoff profiles of such equilibria is the same as the set of all BNE. They
+develop a backward inductive sequential procedure, whose solution, if it exists,
+provides a SPCIB BNE, and identify instances which guarantee the existence
+of SPCIB BNE. The class of dynamic games studied in Tang et al (2022);
+Ouyang et al (2015, 2017); Vasal and Anastasopoulos (2016); Sinha and Anas-
+tasopoulos (2016) satisfy the following assumptions: (i) agents’ actions are
+observable (ii) each agent has a perfect observation of his own local states/-
+type (iii) conditioned on the agents’ actions, the evolution of the local states
+are independent. We relax assumptions (i)-(iii) of Tang et al (2022); Ouyang
+et al (2015, 2017); Vasal and Anastasopoulos (2016); Sinha and Anastasopoulos
+(2016), and study a general class of dynamic games with asymmetric infor-
+mation, hidden actions, imperfect observations, and controlled and coupled
+dynamics.
+1.2 Contribution
+We study/analyze, in discrete time, a general class of sequential stochastic
+dynamic games with asymmetric information, where the underlying system is
+dynamic, the information structure is non-classical, at each time instant the
+agents have private and common information and their actions are hidden
+(each agent’s actions are not directly observable by the other agents). Our key
+contribution is a methodology for the discovery of Bayesian Nash Equilibrium
+(BNE) strategy profiles that are based on the agents’ compressed private and
+
+Springer Nature 2021 LATEX template
+6
+Dynamic Games with Asymmetric Information and Hidden Actions
+common information and can be determined sequentially in time moving back-
+wards, if each step of this backward procedure has a solution. We present an
+example where such a BNE strategy profile exists. We show that our method-
+ology works also for the case where the agents have no common observations
+and their actions are hidden.
+1.3 Organization
+The rest of the paper is organized as follows: We present the game’s model
+along with the equilibrium concept in Section 2. We state our objective and
+present the methodology that achieves it in Section 3. In Section 4 we first
+introduce compressed versions of the agents’ private and common informa-
+tion that are sufficient for decision making purposes; then we define Sufficient
+Information Based (SIB) strategies that are based on the agents’ compressed
+information. In Section 5 we first introduce Sufficient Information Based
+Bayesian Nash Equilibrium (SIB-BNE); then we present a sequential decom-
+position of the game, that is, a backward inductive procedure that determines
+SIB-BNE if each step of this procedure has a solution. In Section 6 we present
+an example that highlights our solution methodology and where a SIB-BNE
+exists. In Section 7 we show that our solution methodology works for stochas-
+tic dynamic games where the agents have no common observations and each
+agent’s actions are part of his private information. The comparison of the
+definitions of compressed private information as it appears in this paper and
+in Tavafoghi et al (2022), along with some of the technical details related to
+the existence of SIB-BNE for the example of Section 6 are presented in the
+Appendices.
+2 Model
+We present our model for dynamic decision problems with strategic agents
+(dynamic games) below; this model is an analogue to the model of Tavafoghi
+et al (2022) for dynamic decision problems with non-strategic agents.
+2.1 System Dynamics
+There are N strategic agents who live in a dynamic Markovian world over
+horizon T := {1, 2, ..., T }, T < ∞. Let Xt ∈ Xt denote the state of the world
+at t ∈ T . At time t, each agent, indexed by i ∈ N := {1, 2, ..., N}, chooses an
+action ai
+t∈Ai
+t, where Ai
+t denotes the set of available actions to him at t. Given
+the collective action profile At := (A1
+t , ..., AN
+t ), the state of the world evolves
+according to the following stochastic dynamic equation,
+Xt+1 = ft(Xt, At, W x
+t ),
+(1)
+where W x
+1:T −1 is a sequence of independent random variables. The initial state
+X1 is a random variable that has a probability distribution µ0 ∈ ∆(X1).
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+7
+At every time t ∈ T , before taking an action, agent i receives a noisy private
+observation Y i
+t ∈ Yi
+t of the current state of the world Xt and the action profile
+At−1, given by
+Y i
+t = Oi
+t(Xt, At−1, W i
+t ),
+(2)
+where W i
+1:T , i ∈ N, are sequences of independent random variables. Moreover,
+at every t ∈ T , all agents receive a common observation Zt ∈ Zt of the current
+state of the world Xt and the action profile At−1, given by
+Zt = Oc
+t(Xt, At−1, W c
+t ),
+(3)
+where W c
+1:T , is a sequence of independent random variables. We assume that
+the random variables X1, W x
+1:T −1, W c
+1:T , and W i
+1:T , i ∈ N are mutually
+independent.
+To avoid measure-theoretic technical difficulties and for clarity and conve-
+nience of exposition, we assume that all the random variables take values in
+finite sets.
+Assumption 1. (finite game) The sets N, Xt, Zt, Yi
+t, Ai
+t, i ∈ N, are finite.
+2.2
+Information Structure
+Let Ht denote the aggregate information of all agents at time t. Assuming that
+agents have perfect recall, we have Ht = {Z1:t, Y 1:N
+1:t , A1:N
+1:t−1}, i.e. Ht denotes
+the set of all agents’ past and present observations and all agents’ past actions.
+The set of all possible realizations of the agents’ aggregate information is given
+by Ht := �
+τ≤t Zτ × �
+i∈N
+�
+τ≤t Yi
+τ × �
+i∈N
+�
+τ<t Ai
+τ.
+At time t∈T , the aggregate information Ht is not fully known to all agents.
+Let Ct := {Z1:t} ∈ Ct denote the agents’ common information about Ht and
+P i
+t := {Y i
+1:t, Ai
+1:t−1}\Ct ∈ Pi
+t denote agent i’s private information about Ht,
+where Pi
+t and Ct denote the set of all possible realizations of agent i’s private
+and common information at time t, respectively. We assume that observations
+Y i
+τ , τ ∈ {1, 2..., t}, and actions Ai
+τ, τ ∈ {1, 2..., t−1}, are known to agent i but
+are not necessarily fully known to all other agents, denoted by −i, at t ∈ T .
+Therefore, we have P i
+t ⊆ {Y i
+1:t, Ai
+1:t−1} for all i ∈ N, and Ht =
+��
+i∈N P i
+t
+�
+∪Ct
+for all t ∈ T . As such,
+�
+Ct, P i
+t , i ∈ N
+�
+form a partition of Ht at every time
+t ∈ T . In Section 2.5, we discuss several instances of information structures
+that can be captured as special cases of our model.
+2.3 Strategies and Utilities:
+Let Hi
+t := {Ct, P i
+t } ∈ Hi
+t denote the information available to agent i at t,
+where Hi
+t denote the set of all possible realizations of agent i’s information at
+t. Agent i’s behavioral strategy at t, denoted by gi
+t, is defined by
+gi
+t : Hi
+t → ∆(Ai
+t)
+(4)
+
+Springer Nature 2021 LATEX template
+8
+Dynamic Games with Asymmetric Information and Hidden Actions
+where ∆(Ai
+t) is the set of Probability Mass Functions (PMFs) on Ai
+t. We
+denote by
+gi := (gi
+1, gi
+2, . . . , gi
+T )
+(5)
+a strategy of agent i; gi ∈ Gi, where Gi is the set of admissible strategies
+described by (4)-(5). We denote a strategy profile g by
+g := (g1, g2, . . . , gN)
+(6)
+g ∈ G, where G is the set of admissible strategy profiles described by (4)-(6).
+We denote by
+g−i := (g1, . . . , gi−1, gi+1, . . . , gN)
+(7)
+Agent i’s instantaneous utility at t depends on the system state Xt and the
+collective action profile At, and is given by ui
+t(Xt,At). Agent i’s total utility
+over horizon T , is given by,
+U i(X1:T , A1:T ) =
+�
+t∈T
+ui
+t(Xt, At).
+(8)
+2.4 Equilibrium Concept:
+We consider Bayesian Nash Equilibrium (BNE) as the solution concept (Fuden-
+berg and Tirole, 1991). A strategy profile g∗ = (g∗1, g∗2, . . . , g∗N) is a BNE if
+for all i ∈ N
+Eg∗{U i(X1:T , A1:T )} ≥ Eg∗−i,ˆgi{U i(X1:T , A1:T )}, ∀ˆgi ∈ Gi.
+(9)
+2.5 Special Cases
+We discuss several instances of dynamic games with asymmetric information
+that are special cases of the general model described above.
+1) Nested information structure: Consider a two-player game with one
+informed player and one uninformed player and general Markovian dynamics.
+At every time t∈T , the informed player makes a private perfect observation of
+the state Xt, i.e. Y 1
+t =Xt. The uninformed player does not have any observa-
+tion of the state Xt. Both the informed and uninformed players observe each
+others’ actions, i.e. Zt={At−1}. Therefore, we have P 1
+t = {X1:t}, P 2
+t = ∅, and
+Ct={A1
+1:t−1,A2
+1:t−1} for all t∈T . The above nested information structure cor-
+responds to dynamic games considered in Renault (2006); Cardaliaguet et al
+(2015); Renault (2012); Li and Shamma (2014, 2017); Zheng and Casta˜n´on
+(2013), where in Renault (2012); Li and Shamma (2017) the state Xt is static.
+2) Delayed sharing information structure: Consider a N-player game with
+observable actions where agents observe each others’ observations with d-step
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+9
+delay. That is, P i
+t = {Y i
+t−d+1:t} and Ct = {Y1:t−d, A1:t−1}. We note that in
+our model we assume that the agents’ common observation Zt at t is only a
+function of Xt and and At−1. Therefore, to describe the game with delayed
+sharing information structure within the context of our model we need to
+augment our state space to include the agents’ last d observations as part of
+the augmented state. Define ˜Xt := {Xt, M 1
+t , M 2
+t , ..., M d
+t } as the augmented
+system state where M i
+t := {At−i, Yt−i} ∈ At−i×Yt−i, i ∈ N; that is, M i
+t serves
+as a temporal memory for the agents’ observation Yt−i at t − i. Then, we have
+˜Xt+1 = {Xt+1, M 1
+t+1, M 2
+t+1, ..., M d
+t+1} = {ft(Xt, At, W x
+t ), (Yt), M 1
+t , ..., M d−1
+t
+}
+and Zt = {M d
+t , At−1} = {Yt−d, At−1}.
+The above environment captures a connection between the symmetric
+information structure and asymmetric information structure. The informa-
+tion asymmetry among the agents increases as d increases. The above delayed
+sharing information structure corresponds to the dynamic game considered in
+Tavafoghi et al (2016).
+3) Perfectly controlled dynamics with hidden actions: Consider a N-player
+game where the state Xt:=(X1
+t,X2
+t,...,XN
+t ) has N components. Agent i, i∈N,
+perfectly controls Xi
+t, i.e. Xi
+t+1 = Ai
+t. Agent i’s actions Ai
+t, t ∈ T , are not
+observable by all other agents −i. Every agent i, i∈N, makes a noisy private
+observation Y t
+i (Xt, W i
+t ) of the system state at t∈T . Therefore, we have P i
+t :=
+{A1:t, Y i
+1:t}, Ct=∅.
+3 Objective and Methodology
+3.1 Objective
+Our objective is twofold: (i) To determine BNE strategy profiles that are based
+on compressed versions of the agents’ private and common information. (ii)
+To compute the above-mentioned strategy profiles by a sequential decomposi-
+tion of the game, that is, by a backward inductive sequential procedure that
+identifies an equilibrium strategy profile when every step of the procedure has
+a solution.
+3.2 Methodology
+We present a methodology that achieves the above-state objective and
+proceeds as follows:
+• Step 1. We determine a mutually consistent compression of the agents’
+private information that is sufficient for decision-making purposes (such a
+mutually consistent compression may not be unique). Based on this com-
+pression we introduce the Sufficient Private Information Based (SPIB)
+belief system.
+• Step 2. Based on the result of Step 1, we determine a compression of the
+agents’ common information that is sufficient for decision-making pur-
+poses by defining the Common Information Based (CIB) belief system.
+The CIB belief system ensures that at each time instant each agent’s CIB
+
+Springer Nature 2021 LATEX template
+10
+Dynamic Games with Asymmetric Information and Hidden Actions
+belief is consistent with his SPIB belief even when the agent deviates from
+his equilibrium strategy and plays an arbitrary strategy. Such a consis-
+tency implies that each agent forms his own CIB belief system, and each
+agent’s CIB belief system is common knowledge among all agents.
+• Step 3. Based on the compression of the agents’ private and common
+information we introduce Sufficient Information Based (SIB) strategies
+for each agent (i.e., strategies that depend at each time on the agent’s suf-
+ficient private information and the CIB belief system) and SIB BNE. We
+show that SIB strategies satisfy a key closedness of best response prop-
+erty. Based on this property we provide a sequential decomposition of the
+game, that is, a backward inductive sequential procedure that determines
+a SIB BNE if each step of the procedure has a solution.
+• Step 4. We provide an example of a stochastic dynamic game with asym-
+metric information and hidden/unobservable actions where a SIB BNE
+exists.
+4 Compression of Private and Common
+Information
+In Section 4.1 we characterize/determine mutually consistent compressions
+of all agents’ private information that are sufficient for decision-making pur-
+poses. In Section 4.2 we introduce the common information based belief, a
+compressed version of the agents’ common information, that is sufficient for
+decision making purposes.
+4.1 Sufficient private information (Step 1)
+We present/consider a compression of the agents’ private information that is
+done in a mutually consistent manner so that the compressed information is
+sufficient for decision making purposes.
+Definition 1 (Sufficient private information). We say that Si
+t, i = 1, . . . , N,
+is sufficient private information for the agents if
+(i) Si
+t is a function of Hi
+t such that Si
+t = ζi
+t(Hi
+t) for some commonly known
+functions ζi
+t, i = 1, 2, . . . , N.
+(ii) Si
+t can be sequentially updated as Si
+t = φi
+t(Si
+t−1, Y i
+t , Zt, Ai
+t−1) using some
+commonly known functions φi
+t, i = 1, 2, . . . , N.
+(iii) For any realization xt, p−i
+t , pi
+t, ct, and the corresponding s−i
+t
+= ζ−i
+t (p−i
+t , ct)
+and si
+t = ζi
+t(pi
+t, ct), and any strategy profile g, where gi
+t : Si
+t × Ct →
+∆(Ai
+t), ∀i, ∀t, such that Pg(pi
+t, ct) > 0,
+Pg(xt, s−i
+t
+| si
+t, ct) = Pg(xt, s−i
+t
+| pi
+t, ct)
+(10)
+Remark 1. A similar definition of sufficient private information for dynamic
+teams appears in (Tavafoghi et al, 2022, Definition 2). This definition is slightly
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+11
+different from Definition 1 above because the objectives in Tavafoghi et al
+(2022) and this paper are different. In Appendix .1 we show that sufficient
+private information satisfying Definition 1 may violate condition (ii) of Defi-
+nition 2 in Tavafoghi et al (2022). In Tavafoghi et al (2022) the compression
+of private (and common) information must entail no loss in performance, that
+is, we must be able to determine globally optimal team strategy profiles that are
+based on compressed private and common information. In this paper the goal
+is to determine BNE strategy profiles that are based on compressed informa-
+tion and be sequentially computed (if such BNE strategy profiles exist). We are
+not concerned about the equilibria we may lose when we compress information;
+therefore, we don’t need condition (ii) of Definition 2 in Tavafoghi et al (2022).
+Definition 1 characterizes a set of compressions for agents’ private infor-
+mation. In the following, we show the set of sufficient private information Si
+t,
+i ∈ N, t ∈ N, is rich enough to form belief systems on information sets of
+realizations with positive or zero probability. Let ˜gi denote the uniform strat-
+egy that assigns equal probability to every action of agent i ∈ N. Below we
+show that the policy-independence property of belief (Tavafoghi et al, 2022,
+Theorem 1) for agent i is still true when the private information pi
+t is replaced
+with the sufficient private information si
+t. That is, P˜gi,g−i(xt, x−i
+t
+| si
+t, ct) con-
+structed by (˜gi, g−i) captures agent i’s belief based on hi
+t even when he plays
+an arbitrary strategy ˆgi, not necessarily the same as gi or ˜gi, provided that
+agents −i play g−i.
+Lemma 1. For hi
+t such that Pˆgi,g−i(hi
+t) > 0, we have P˜gi,g−i(hi
+t) > 0 and
+Pˆgi,g−i(xt, s−i
+t
+| hi
+t) = P˜gi,g−i(xt, s−i
+t
+| hi
+t) = P˜gi,g−i(xt, s−i
+t
+| si
+t, ct).
+(11)
+Proof Note that P˜gi(ai
+t) = 1/|Ai
+t|, so P˜gi,g−i(hi
+t) > 0 given that Pg(hi
+t) > 0. Then
+from part (i) of the definition of sufficient private information and part (i) of Theorem
+1 in Tavafoghi et al (2022) we have
+Pˆgi,g−i
+(xt, s−i
+t
+| hi
+t) =
+�
+h−i
+t
+:ζ−i
+t
+(h−i
+t
+)=s−i
+t
+Pˆgi,g−i
+(xt, h−i
+t
+| hi
+t)
+=
+�
+h−i
+t
+:ζ−i
+t
+(h−i
+t
+)=s−i
+t
+P˜gi,g−i
+(xt, h−i
+t
+| hi
+t)
+= P˜gi,g−i
+(xt, s−i
+t
+| hi
+t).
+(12)
+Furthermore, from condition (iii) of the definition of sufficient private information
+we have
+P˜gi,g−i
+(xt, s−i
+t
+| hi
+t) = P˜gi,g−i
+(xt, s−i
+t
+| si
+t, ct).
+(13)
+□
+
+Springer Nature 2021 LATEX template
+12
+Dynamic Games with Asymmetric Information and Hidden Actions
+4.2 CIB Belief System (Step 2)
+Given the compressed private information, we next compress the agents’ com-
+mon information in the form of a belief system. We call such a compressed
+belief system the Common Information Based (CIB) belief system. Similar to
+Tang et al (2022); Ouyang et al (2017), the CIB belief system is sufficient
+for decision-making if it is common knowledge among all agents, and every
+agent i can compute his belief about the system state and the other agents’
+sufficient private information using the CIB belief system and his compressed
+private information. More specifically, agent i should be able to compute
+Pˆgi,g−i(xt, st | hi
+t) using the CIB belief system and his sufficient private infor-
+mation si
+t whenever other agents follow the strategy profile g−i and agent i
+plays an arbitrary strategy ˆgi.
+To determine a CIB belief system that satisfies the above sufficiency
+requirement we proceed as follows. We first define N CIB belief systems Πψ :=
+{Πψ,1, Πψ,2, . . . , Πψ,N}, one for each agent (Definition 2 below). Each belief
+system Πψ,i consists of a sequence of PMFs on Xt × St that are sequentially
+updated according to an update rule ψ = (ψ1, ψ2, . . . , ψN) that is common
+knowledge among the agents; for each realization ct of the common information
+available at t, πψ,i
+t
+describes the belief on Xt×St based on ct from agent i’s point
+of view. We want πψ,i
+t
+, combined with si
+t, to enable agent i to form his own suf-
+ficient information-based private belief (given by Pˆgi,g∗−i(xt, st | si
+t, ct)) about
+the current status of the game. Furthermore, we want the CIB belief system
+to capture the current status of the game when agents utilize strategies based
+on (St, Πψ
+t ). For that matter, we define the notion/concept of Sufficient Infor-
+mation Based (SIB) strategy profile σ := (σi, i ∈ N), σi := (σi
+t, t ∈ T ), i ∈ N.
+Each component σi
+t of σ is a function of si
+t, agent i’s sufficient private infor-
+mation at t, and πψ
+t = (πψ,i
+t
+, i ∈ N) (see Definition 3 below). Using the N
+CIB belief systems and the SIB strategy profile σ we define update equations
+for each πψ,i
+t
+so that each πψ,i
+t
+is consistent with si
+t and with agent i’s suffi-
+cient private information-based belief Pˆgi,g∗−i(xt, st | si
+t, ct), defined in Section
+4.1 (Definition 1), and each πψ,i
+t
+is common knowledge among all agents (see
+Definition 4 below). We proceed with the (formal) definitions.
+Definition 2 (Common information based (CIB) belief system). Given a
+sequence of update functions ψ = {ψi
+t, i ∈ N, t ∈ T } that are common
+knowledge among the N agents, sequentially define
+Πψ,i
+t
+= ψi
+t(Πψ
+t−1, Zt), i ∈ N, t ∈ T
+(14)
+where
+Πψ
+t :=
+
+
+Πψ,1
+t
+...
+Πψ,N
+t
+
+ , t ∈ T
+(15)
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+13
+Πψ
+0 :=
+
+
+µ0
+...
+µ0
+
+
+(16)
+The sequence Πψ
+1:T = (Πψ
+1 , Πψ
+2 , . . . , Πψ
+T ) defines a CIB belief system; Πψ,i
+t
+denotes the CIB belief over Xt × St based on Ct from agent i’s point of view.
+Definition 3 (SIB strategy). Given a CIB belief system Πψ
+1:T , we define a
+Sufficient Information Based (SIB) strategy profile σ := (σ1, σ2, . . . , σN), σi :=
+(σi
+1, σi
+2, . . . , σi
+T ) by the maps
+σi
+t : Si
+t × [∆(Xt × St)]N → ∆(At), t = 1, 2, . . . , i = 1, 2, . . . , N.
+(17)
+Based on Definitions 2 and 3 we present a set of conditions that an indi-
+vidual CIB belief system (Πψ,i
+t
+, t ∈ T ) must satisfy so as to ensure that each
+agent i can form his own (private) belief about the current status of the game,
+given by (Xt, St), using Πψ
+t and Si
+t when all other agents −i employ SIB strate-
+gies σ−i. This set of conditions describe a sequential update rule of Πψ,i
+t
+; the
+update rule depends on whether or not the (new) common observation at t is
+feasible under the agents’ strategies.
+Definition 4 (Consistent CIB belief system). Consider a SIB strategy
+profile σ. Let F i
+t (xt+1, st+1, zt+1)(πψ
+t ;
+σ−i
+t ) denote the CIB belief about
+(xt+1, st+1, zt+1) constructed recursively by assuming that (i) (xt, st) is dis-
+tributed according to πψ,i
+t
+(ii) agent i employs the uniform strategy ˜gi at t (i.e.,
+the strategy that chooses every action ai
+t ∈ Ai
+t with equal probability), and (iii)
+agent −i plays according σ−i
+t . That is,
+F i
+0(x1, s1, z1) =
+�
+y1
+�
+P{z1, y1 | x1}µ0(x1)
+��
+j
+1{sj
+1 = φj
+1(z1, yj
+1)}
+� �
+(18)
+at t = 1, and for t ≥ 1.
+F i
+t (xt+1, st+1, zt+1)(πψ
+t ; σ−i
+t )
+=
+�
+yt+1,xt,st,at
+�
+P{zt+1, yt+1, xt+1 | xt, at}
+��
+j
+1{sj
+t+1 = φj
+t+1(sj
+t, zt+1, yj
+t+1, aj
+t)}
+�
+
+
+1
+| Ai
+t |
+�
+j̸=i
+σj
+t (aj
+t)(πψ
+t , sj
+t)
+
+ πψ,i
+t
+(xt, st)
+�
+(19)
+We define the update rule ψσ = (ψσ,i
+t , i ∈ N, t ∈ T ) and the corresponding
+CIB belief system Πψσ
+1:T as follows. At any t
+
+Springer Nature 2021 LATEX template
+14
+Dynamic Games with Asymmetric Information and Hidden Actions
+(i) If �
+ˆxt+1,ˆst+1 F i
+t (ˆxt+1, ˆst+1, zt+1)(πψσ
+t
+; σ−i
+t ) > 0 (i.e. the new common
+observation zt+1 is feasible from the agent i’s point of view), then πψσ,i
+t+1
+can be updated recursively as
+πψσ,i
+t+1 (xt+1, st+1) =
+F i
+t (xt+1, st+1, zt+1)(πψσ
+t
+; σ−i
+t )
+�
+ˆxt+1,ˆst+1 F i
+t (ˆxt+1, ˆst+1, zt+1)(πψσ
+t
+; σ−i
+t )
+,
+(20)
+via Bayes rule.
+(ii) If �
+ˆxt+1,ˆst+1 F i
+t (ˆxt+1, ˆst+1, zt+1)(πψσ
+t
+; σ−i
+t ) = 0 (i.e. the new common
+observation zt+1 is infeasible from the agent i’s point of view), then the
+update rule is
+πψσ,i
+t+1 (xt+1, st+1) =
+1
+|Xt+1 × St+1|.
+(21)
+Based on (20) and (21) we can write
+Πψσ,i
+t+1 = ψσ,i
+t+1(Πψσ
+t , Zt+1).
+(22)
+Πψσ
+t+1 = ψσ
+t+1(Πψσ
+t , Zt+1).
+(23)
+Furthermore,
+for
+all
+i
+∈
+N,
+each
+agent
+can
+determine
+if
+�
+ˆxt+1,ˆst+1 F i
+t (ˆxt+1, ˆst+1, zt+1)(πσψ
+t
+; σ−i
+t ) is positive or zero; thus each agent
+knows how agent i computes πψσ,i
+t+1 from σi
+t, zt+1, σ−i
+t
+and ψσ. Therefore, πψσ,i
+t
+(hence πψσ
+t
+) is common knowledge among all agents. We call Πψσ
+1:T the CIB
+belief system consistent with the SIB strategy profile σ.
+Remark 2. Since the sufficient private information is a function of the agent’s
+available information, a SIB strategy σi
+t corresponds to a strategy gi,σ
+t
+given
+by gi,σ
+t (hi
+t) := σi
+t(ζi
+t(hi
+t), πψσ
+t
+). Therefore, in the rest of the paper we use the
+following convention: Pσ(·) = Pgσ(·) and Eσ[·] = Egσ[·].
+Remark 3. There are many alternative specifications of the update rule
+ψσ
+t , t ∈ T defined by (22)-(23), that result in consistent CIB belief systems,
+that is, CIB belief systems which ensure that (i) agent i can form his pri-
+vate belief over (Xt, S−i
+t ) by incorporating his private sufficient information
+Si
+t into his CIB belief Πψσ,i
+t
+given that agents −i play according to σ−i,
+(ii) agent i’s private belief formed according to i is identical to the prob-
+ability distribution over (Xt, S−i
+t ) conditional on his complete history Hi
+t
+even when he plays an arbitrary strategy ˆgi different from σi. An example
+of such an alternative update rule is described by (20) (Bayes’ rule) when
+�
+ˆxt+1,ˆst+1 F i
+t (ˆxt+1, ˆst+1, zt+1)(πψσ
+t
+; σ−i
+t ) > 0 and a arbitrary PMF πψσ,i
+t+1 (·, ·)
+on Xt+1 × St+1 when �
+ˆxt+1,ˆst+1 F i
+t (ˆxt+1, ˆst+1, zt+1)(πψσ
+t
+; σ−i
+t ) = 0.
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+15
+Definition 4 ensures that agent i can form his beliefs over (Xt, S−i
+t ) by
+incorporating his sufficient private information Si
+t into his CIB belief Πψσ,i
+t
+given that agents −i play according to σ−i. Moreover, this belief is sufficient to
+compute the probability distribution over (Xt, S−i
+t ) conditional on his complete
+history Hi
+t even when he plays an arbitrary strategy ˆgi different from σi.
+We formalize the above discussion in Lemma 2 below, by using the notation
+Pˆgi,σ−i,ψσ(·) to indicate the belief resulting when agent i plays ˆgi and agents
+−i play g−i,σ(h−i
+t ) = σ−i
+t (ζ−i
+t (h−i
+t ), πψσ
+t
+) using the update rule ψσ .
+Lemma 2. Consider a SIB strategy profile σ, along with an associated consis-
+tent CIB belief system Πψσ
+t . Suppose (xt, hi
+t, h−i
+t
+is a realization with positive
+probability under (ˆgi, σ−i), where ˆgi denotes an arbitrary strategy for agent
+i. Let si
+t = ζi
+t(hi
+t) and s−i
+t
+= ζ−i
+t (h−i
+t ) be the associated sufficient private
+information. Then agent i’s belief at time t can be computed using πψσ
+t
+as
+Pˆgi,σ−i,ψσ(xt, s−i
+t
+| hi
+t) =
+πψσ,i
+t
+(xt, st)
+�
+s−i
+t
+,xt πψσ,i
+t
+(xt, si
+t, s−i
+t )
+(24)
+Proof From Lemma 1 we have
+Pˆgi,σ−i,ψσ
+(xt, s−i
+t
+| hi
+t) = P˜gi,σ−i,ψσ
+(xt, s−i
+t
+| hi
+t). = P˜gi,σ−i,ψσ
+(xt, s−i
+t
+| ct, si
+t).
+(25)
+By Bayes’ rule we obtain
+P˜gi,σ−i,ψσ
+(xt, s−i
+t
+| ct, si
+t) = P˜gi,σ−i,ψσ(xt, st | ct)
+P˜gi,σ−i,ψσ(si
+t | ct)
+=
+πψσ,i
+t
+(xt, st)
+�
+s−i
+t
+,xt πψσ,i
+t
+(xt, si
+t, s−i
+t )
+.
+(26)
+Combination of (25) and (26) establishes the assertion of Lemma 2.
+□
+Remark 4. Suppose Xt = (X1
+t , X2
+t , . . . . , XN
+t ) and we have the conditional
+independence property, namely, that for any strategy profile g Pg(xt, st | ct) =
+�
+i Pgi(xi
+t, si
+t | ct). Then one can show for any i that
+πψσ,i
+t
+(xt, st) =
+�
+j
+πψσ,i(xj
+t, sj
+t) = P˜gi
+t(xi
+t, si
+t | ct)
+�
+j̸=i
+Pσj(xj
+t, sj
+t | ct)
+Therefore, for settings with the conditional independence property as in Tang
+et al (2022); Ouyang et al (2017), one can use the simplified beliefs P˜gi
+t(xi
+t, si
+t |
+ct) and Pσj(xj
+t, sj
+t | ct) as the compressed common information to compute
+the CIB belief πψσ,i
+t
+(xt, st). The conditional independence among the system
+components in the models of Tang et al (2022); Ouyang et al (2017) could be
+lost when the agents’ actions are not observable.
+
+Springer Nature 2021 LATEX template
+16
+Dynamic Games with Asymmetric Information and Hidden Actions
+5
+Sequential decomposition (Step 3)
+In this section we present a sequential decomposition of the game, that is, a
+backward inductive sequential procedure that determines a Sufficient Informa-
+tion Based Bayesian Nash Equilibrium (SIB-BNE), defined below, if each step
+of this procedure has a solution. We proceed as follows. We first establish a
+key closedness of best response property (Section 5.1); we use this property to
+provide a sequential decomposition of the game (Section 5.2)
+Definition
+5
+(SIB-BNE).
+Consider
+a
+SIB
+strategy
+profile
+σ∗
+=
+(σ∗1, σ∗2, . . . , σ∗n) and its corresponding consistent update rule ψσ∗. The SIB
+strategy profile σ∗ is a SIB-BNE if it is a BNE of the dynamic game. That is,
+for all i ∈ N,
+Eˆgi,σ∗−i,ψσ∗
+{U i(X1:T , A1:T )} ≤ Eσ∗,ψσ∗
+{U i(X1:T , A1:T )},
+for all strategies (not necessarily SIB strategies) ˆgi.
+(27)
+5.1 Closedness of best response
+The key result of this subsection is presented in the following theorem.
+Theorem 1. Consider a fixed and known SIB strategy profile σ and the cor-
+responding update rule ψσ. Suppose agents −i use σ−i with ψσ. Then, there
+exists a SIB strategy ˆσi that uses ψσ and is a best response to σ−i with ψσ.
+The proof is based on Lemmas 3, 4, and 5 that we state and prove below.
+Lemma 3. Consider a SIB strategy profile σ and the corresponding update
+rule ψσ along with the consistent CIB belief system Πψσ
+1:T .
+If agents −i play according to the SIB strategies σ−i and use the update
+rule ψσ, the best response problem for agent i is a POMDP with state and
+observation processes
+˜Xt = (St, Πψσ
+t , Xt), t ∈ T
+(28)
+˜Yt = (Y i
+t , Zt), t ∈ T
+(29)
+respectively, and instantaneous utility
+˜ui
+t( ˜Xt, Ai
+t) =
+�
+a−i
+t
+� �
+j̸=i
+σj
+t (aj
+t | Sj
+t , Πψσ
+t )
+�
+ui
+t(Xt, a−i
+t , Ai
+t), t ∈ T
+(30)
+The assertion of Lemma 3 is a direct consequence of Lemmas 4 and 5.
+Lemma 4. Consider a SIB strategy profile σ and the corresponding update
+rule ψσ. Suppose agents −i play according to the SIB strategies σ−i using ψσ
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+17
+and agent i follows an arbitrary strategy ˆgi (not necessarily a SIB strategy).
+Then
+Pˆgi,σ−i,ψσ(˜xt+1, ˜yt+1 | ˜x1:t, ˜y1:t, ai
+1:t) = Pˆgiσ−i,ψσ(˜xt+1, ˜yt+1 | ˜xt, ai
+t)
+(31)
+Proof The probability for the next state and observation ˜xt+1, ˜yt+1 can be computed
+by
+Pˆgi,σ−i,ψσ
+(˜xt+1, ˜yt+1 | ˜x1:t, ˜y1:t, ai
+1:t)
+= Pˆgi,σ−i,ψσ
+(xt+1, πψσ
+t+1, st+1, yi
+t+1, zt+1 | x1:t, πψσ
+1:t , s1:t, yi
+1:t, z1:t, ai
+1:t)
+=
+�
+y−i
+t+1,a−i
+t
+Pˆgi,σ−i,ψσ
+(xt+1, πψσ
+t+1, st+1, yt+1, zt+1, a−i
+t
+| x1:t, πψσ
+1:t , s1:t, yi
+1:t, z1:t, ai
+1:t)
+=
+�
+y−i
+t+1,a−i
+t
+� �
+j
+1(sj
+t+1 = φj
+t+1(sj
+t, yj
+t+1, zt+1, aj
+t))
+�
+P{zt+1, yt+1, xt+1 | xt, at}
+1(πψσ
+t+1 = ψσ
+t+1(πψσ
+t
+, zt+1))
+� �
+j̸=i
+σj
+t (aj
+t | sj
+t, πψσ
+t
+)
+�
+(32)
+where the last equality follows from the system dynamics, part (ii) of Definition 1,
+Definition 4, and the form of SIB strategies of agents −i. Since the right hand side
+of (32) depends only on (˜xt, ai
+t) we conclude that
+Pˆgi,σ−i,ψσ
+(˜xt+1, ˜yt+1 | ˜x1:t, ˜y1:t, ai
+1:t) = Pˆgi,σ−i,ψσ
+(˜xt+1, ˜yt+1 | ˜xt, ai
+t)
+(33)
+□
+Lemma 4 shows that { ˜Xt, ˜Yt, t ∈ T } is a Markov process conditional on
+{Ai
+t, t ∈ T }
+Lemma 5. Consider a SIB strategy profile σ and the corresponding update
+rule ψσ. Suppose agents −i follow the SIB strategies σ−i using ψσ and agent i
+follows an arbitrary strategy ˆgi (not necessarily a SIB strategy). Then there are
+utility functions ˜ui
+t such that Eˆgi,σ−i,ψσ[˜ui
+t( ˜Xt, Ai
+t)] = Eˆgi,σ−i,ψσ[ui
+t(Xt, At)]
+for all t ∈ T .
+Proof Recall that ˜
+Xt = (St, Πψσ
+t
+, Xt). Then
+Eˆgi,σ−i,ψσ
+[ui
+t(Xt, At)]
+= Eˆgi,σ−i,ψσ
+[ui
+t(Xt, A−i
+t , Ai
+t)]
+= Eˆgi,σ−i,ψσ �
+Eˆgi,σ−i,ψσ
+[ui
+t(Xt, A−i
+t , Ai
+t) | ˜
+Xt, Ai
+t]
+�
+= Eˆgi,σ−i,ψσ � �
+a−i
+t
+Pˆgi,σ−i,ψσ
+(a−i
+t
+| St, Πψσ
+t
+, Xt, Ai
+t)ui
+t(Xt, a−i
+t , Ai
+t)]
+�
+= Eˆgi,σ−i,ψσ � �
+a−i
+t
+� �
+j̸=i
+σj
+t (aj
+t | Sj
+t , Πψσ
+t
+)
+�
+ui
+t(Xt, a−i
+t , Ai
+t)]
+�
+(34)
+
+Springer Nature 2021 LATEX template
+18
+Dynamic Games with Asymmetric Information and Hidden Actions
+Therefore, we establish the claim of the lemma by defining
+˜ui
+t( ˜Xt, Ai
+t) =
+�
+a−i
+t
+� �
+j̸=i
+σj
+t (aj
+t | Sj
+t , Πψσ
+t
+)
+�
+ui
+t(Xt, a−i
+t , Ai
+t)]
+(35)
+□
+Proof of Theorem 1 From Lemma 3 we conclude that the best response of agent
+i to σ−i is a POMDP with state ˜Xt. From the theory of POMDP (Kumar and
+Varaiya, 1986, Chapter 6) we know that: (i) the belief on the state ˜
+Xt = (St, Πψσ
+t
+, Xt)
+conditioned on available information hi
+t is an information state for the agent; (ii)
+for each t ∈ T there exists an optimal strategy for agent i that is a function of the
+information state at t. We now prove that (Si
+t, Πψσ
+t
+) is an information state for agent
+i at t, t ∈ T .
+We note that Si
+t+1 = φi
+t(Si
+t, Y i
+t+1, Zt+1, Ai
+t) from part (ii) of Definition 1, and Πψσ
+t+1 =
+ψσ
+t+1(Πψσ
+t
+, Zt+1) from (23).
+Thus, we only need to show that for any strategy ˆgi and any realization hi
+t such
+that Pˆgi,σ−i,ψσ(hi
+t) > 0 the following equality is true:
+Pˆgi,σ−i,ψσ
+(st, πψσ
+t
+, xt | hi
+t) = Pˆgi,σ−i,ψσ
+(st, πψσ
+t
+, xt | si
+t, πψσ
+t
+)
+(36)
+For that matter, we note that si
+t, πψσ
+t
+are perfectly known to agent i. Furthermore,
+from the definition of sufficient private information and Lemma 2 we have
+Pˆgi,σ−i,ψσ
+(s−i
+t , xt | hi
+t) =
+πψσ,i
+t
+(st, xt)
+�
+s−i
+t
+,xt πψσ,i
+t
+(si
+t, s−i
+t , xt)
+,
+(37)
+which is a function of (si
+t, πψσ
+t
+). Therefore,
+Pˆgi,σ−i,ψσ
+(st, πψσ
+t
+, xt | hi
+t) = 1(si
+t = ζi
+t(hi
+t))1(πψσ
+t
+= γψσ
+(hi
+t)) Pˆgi,σ−i,ψσ
+(s−i
+t , xt | pi
+t, ct)
+(38)
+where γψσ(hi
+t) = ψσ
+t (ψσ
+t−1, · · · ) is the composition of ψσ from 1 to t. Then, equation
+(36) is true because of (37) and (38). Consequently, (Si
+t, Πψσ
+t
+), t ∈ T is an information
+state for the best response problem for agent i and the assertion of Theorem 1 is
+true.
+□
+As a result of Theorem 1, a definition of SIB BNE equivalent to Definition
+5 is the following
+Definition 6 (Equivalent definition of SIB BNE). Consider a SIB strategy
+profile σ∗ = (σ∗1, σ∗2, . . . , σ∗n) and its corresponding consistent update rule
+ψσ∗. The SIB strategy profile σ∗ is a SIB BNE if for all i ∈ N,
+Eσi,σ∗−i,ψσ∗
+{U i(X1:T , A1:T )} ≤ Eσ∗,ψσ∗
+{U i(X1:T , A1:T )}
+(39)
+for all σi ∈ Λi where Λi is the set of SIB strategy profiles of agent i.
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+19
+A consequence of Lemmas 3-5 and Theorem 1 is the following. Consider
+a SIB strategy profile σ, the corresponding update rule ψσ along with the
+consistent CIB belief system Πψσ
+1:T ; if agents −i play according to σ−i, then the
+best response of agent i could be determined by the dynamic program
+˘V i
+T +1(·, ·) = 0 for all i
+(40)
+˘V i
+t (πψσ
+t
+, si
+t) = max
+˜σi
+t∈Λi
+t
+E˜σi
+t,σ−i
+t
+,ψσ{ui
+t(Xt, At) + ˘V i
+t+1(ψσ
+t+1(πψσ
+t
+, Zt+1), Si
+t+1) | si
+t},
+∀πψσ
+t
+∈ ∆(Xt × St)N, ∀si
+t ∈ Si
+s, t ∈ T
+(41)
+where Λi
+t is the set of SIB strategies of agent i at time t.
+5.2 Sequential decomposition
+Given a set of value functions Vt+1 = {V i
+t+1 : Πt+1 × Si
+t+1 → R, i ∈ N}, a SIB
+strategy profile σ, the corresponding update rule ψσ
+t+1 defined by (23), and the
+consistent CIB belief πψσ
+t
+, define the stage-game Gt(Vt+1, πψσ
+t
+) as follows.
+(i) There are N agents. (ii) The system state is Xt. (iii) Each agent i
+observes private information Si
+t and common information πψσ
+t
+. (iv) Agent i’s
+belief about the state Xt and other agents’ private information S−i
+t
+is given
+by πψσ,i
+t
+(xt, s−i
+t ), that is,
+πψσ,i
+t
+(xt, s−i
+t ) ∈ ∆(Xt × S−i
+t ).
+(42)
+(v) Each agent i selects action Ai
+t based on his available information; let ˆσi
+t
+denote agent i’s strategy for this stage-game; then,
+Pˆσt,ψσ(Ai
+t = ai
+t | si
+t, πψσ
+t
+) = ˆσi
+t(ai
+t | si
+t, πψσ
+t
+).
+(43)
+(vi) Each agent i has utility
+U i
+Gt(Vt+1,πψσ
+t
+) = ui
+t(Xt, At) + V i
+t+1(ψσ
+t+1(πψσ
+t
+, Zt+1), Si
+t+1)
+(44)
+where (Zt+1, Si
+t+1) conditioned on (Xt, St, At) follows the conditional proba-
+bility �
+xt+1,s−i
+t+1 P(zt+1, xt+1, st+1 | xt, st, at) and the conditional probability
+P(zt+1, xt+1, st+1 | xt, st, at) is given by
+P(zt+1, xt+1, st+1 | xt, st, at)
+=
+�
+yt+1
+P{xt+1 | xt, at}P{zt+1, yt+1 | xt+1, at}
+��
+j
+1{sj
+t+1 = φj
+t+1(sj
+t, zt+1, yj
+t+1, aj
+t)}
+�
+(45)
+
+Springer Nature 2021 LATEX template
+20
+Dynamic Games with Asymmetric Information and Hidden Actions
+(vii) Given a strategy profile ˆσt for the stage-game, the expected utility of
+each player i is given by
+Eˆσt,ψσ[U i
+Gt(Vt+1,πψσ
+t
+) | si
+t]
+=
+�
+xt,s−i
+t
+,at,zt+1,xt+1,st+1
+πψσ,i
+t
+(xt, s−i
+t )
+�
+j
+ˆσj
+t (ai
+t | si
+t, πψσ
+t
+) P(zt+1, xt+1, st+1 | xt, st, at)
+(ui
+t(xt, at) + V i
+t+1(ψσ
+t+1(πψσ
+t
+, zt+1), si
+t+1))
+(46)
+Note that all the random variables of the stage-game Gt(Vt+1, πψσ
+t
+) may
+not necessarily be the same as their counterparts in the original dynamic game
+since each agent i is allowed to choose an arbitrary SIB strategy ˆσi
+t which may
+be different from σi
+t specified by the SIB strategy profile σ. The stage-game
+random variables will coincide with their counterparts in the original game if
+all agents follow σ.
+Theorem 2 (Sequential decomposition). Consider a SIB strategy profile σ =
+{σt, t ∈ T } and the corresponding update rule ψσ = {ψσ
+t , t ∈ T } defined by
+(22)-(23). Define
+V i
+T +1(·, ·) = 0 for all i
+(47)
+V i
+t (πψσ
+t , si
+t) = Eσt,ψσ[U i
+Gt(Vt+1,πψσ
+t
+) | si
+t]
+(48)
+where the right hand side of (48) is given by (46). If for all t ∈ T , there is a
+SIB strategy profile ˆσt such that ˆσt is a BNE of the stage-game Gt(Vt+1, πψσ
+t
+),
+that is,
+Eˆσi
+t,ˆσ−i
+t
+,ψσ[U i
+Gt(Vt+1,πψσ
+t
+) | si
+t] = max
+˜σi
+t∈Λi
+t
+E˜σi
+t,ˆσ−i
+t
+,ψσ[U i
+Gt(Vt+1,πψσ
+t
+) | si
+t]
+(49)
+for all i ∈ N where Λi
+t is the set of SIB strategies of agent i at time t, and
+ˆσt = σt,
+(50)
+then the SIB strategy profile σ is a SIB-BNE of the original dynamic game.
+Proof Suppose that for all t ∈ T there is a SIB strategy profile ˆσt = (ˆσ1
+t , ˆσ2
+t , . . . , ˆσN
+t )
+that is a BNE of the stage game Gt(Vt+1, πψσ
+t
+). Then for all πψσ
+t
+∈ ∆(Xt×St)N, si
+t ∈
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+21
+Sis
+Eˆσi
+t,ˆσ−i
+t
+,ψσ
+[Ui
+Gt(Vt+1,πψσ
+t
+) | si
+t]
+= max
+˜σi
+t∈Λi
+t
+E˜σi
+t,ˆσ−i
+t
+,ψσ
+[ui
+t(Xt, At) + V i
+t+1(ψσ
+t+1(πψσ
+t
+, Zt+1), Si
+t+1) | si
+t].
+(51)
+Equation (51) holds for all t ∈ T with V i
+T +1(·, ·) = 0 and for all i ∈ N. When ˆσt = σt
+for all t ∈ T , Equation (51) gives, for all πψσ
+t
+∈ ∆(Xt × St)N, si
+t ∈ Sis,
+V i
+t (πψσ
+t
+, si
+t) = Eσi
+t,σ−i
+t
+,ψσ
+[Ui
+Gt(Vt+1,πψσ
+t
+) | si
+t]
+= max
+˜σi
+t∈Λi
+t
+E˜σi
+t,σ−i
+t
+,ψσ
+[ui
+t(Xt, At) + V i
+t+1(ψσ
+t+1(πψσ
+t
+, Zt+1), Si
+t+1) | si
+t]
+(52)
+for all i ∈ N.
+By induction, (52), and the fact that the update rule ψσ is consistent with σ we
+have, for all i ∈ N and t ∈ T ,
+E˜σi
+t:T ,σ−i
+t:T ,ψσ
+[
+T
+�
+τ=t
+ui
+τ (Xτ , Aτ) | si
+τ] ≤ Eσi
+t:T ,σ−i
+t:T ,ψσ
+[
+T
+�
+τ=t
+ui
+τ (Xτ , Aτ) | si
+τ]
+(53)
+Then (53) at time t = 1 gives
+E˜σi,σ−i,ψσ
+{Ui(X1:T , A1:T )} ≤ Eσ,ψσ
+{Ui(X1:T , A1:T )}
+(54)
+for all ˜σi ∈ Λi for all i ∈ N. Therefore, the strategy profile σ is a SIB-BNE of the
+original dynamic game (sf. Definition 6).
+□
+Remark 5. Note that even when the stage-game Gt(Vt+1, πψσ
+t
+) has a BNE
+ˆσt, it is possible that ˆσt ̸= σt. Thus, the existence of BNE for every stage-
+game Gt(Vt+1, πψσ
+t
+) is not sufficient to establish the existence of BNE for the
+original dynamic game.
+Remark 6. In the model of Tang et al (2022) when each team consists of
+one agent, a SIB BNE coincides with a SPCIB BNE introduced in Tang et al
+(2022) with an appropriate mapping of the information state as discussed in
+Remark 4.
+Remark 7. There may not be a solution for the set of value functions in the
+sequential decomposition equations described by (47)-(50) for all i ∈ N and
+for all t ∈ T .
+Remark 8. In Definition 4, (21) could be defined differently, and different
+(21) would lead to different choices of ψ. And for any choice of (21), the claim
+of Theorem 2 will still hold.
+Remark 9. The value functions of the sequential decomposition equations
+defined by Theorem 2 (Eqs. (47)-(50) for all i ∈ N, t ∈ T ) may not be
+continuous in the CIB belief Πψσ
+t .
+
+Springer Nature 2021 LATEX template
+22
+Dynamic Games with Asymmetric Information and Hidden Actions
+6 An illustrative example (Step 4)
+In Section 5 we argued (cf. Remark 7) that the sequential decomposition
+equations defined by (47)-(50) for all i ∈ N, t ∈ T may not have a solution,
+and that the value functions defined by (47)-(50) may not be continuous in
+the CIB belief Πψσ
+t
+(cf. Remark 9). In this section we present an example that
+illustrates/highlights the above remarks. In the example, a two-stage stochas-
+tic dynamic game, the agents’ utilities depend on a parameter c. We show
+that: (i) the value functions of the corresponding sequential decomposition
+equations are not continuous in the CIB belief Πψσ
+t ; (ii) for certain values of c
+a SIB-BNE exists.
+6.1 Model
+We consider the following two-stage stochastic dynamic game. There are two
+players/agents, Alice and Bob. At stage one, t = 1, the system’s state X1
+is distributed on {−1, 1} with µ0(−1) = P(X1 = −1) = 0.5 and µ1(1) =
+P(X1 = 1) = 0.5. Alice observes perfectly X1, i.e., Y Alice
+1
+= X1, and takes
+action AAlice
+1
+∈ {−1, 1}; AAlice
+1
+is not observable by Bob and Y Bob
+1
+= ∅. Bob
+does not act at t = 1. At stage 2, t = 2, the system state is X2 = X1AAlice
+1
+.
+Alice and Bob have a common observation Z2 = X2AAlice
+1
+W1 = X1W1, where
+W1 ∈ {−1, 1} and P(Z = i | X1 = i) = 1 − p = 0.8, i ∈ {−1, 1}, and there are
+no private observations, i.e., Y Alice
+2
+= Y Bob
+2
+= ∅. Here p = 0.2 = P(W1 = −1).
+Bob acts at t = 2. Alice does not act at t = 2. Bob’s action ABob
+2
+∈ {−1, 1}.
+Alice’s payoffs at t = 1 and t = 2 are
+uAlice
+1
+(X1, A1) =
+�
+c
+if AAlice
+1
+= 1
+0 if AAlice
+1
+= −1
+(55)
+and
+uAlice
+2
+(X2, A2) =
+
+
+
+2 if X2 = 1, ABob
+2
+= 1
+1 if X2 = −1, ABob
+2
+= −1
+0 otherwise
+(56)
+respectively. Bob’s payoffs are uBob
+t
+(Xt, At) = −uAlice
+t
+(Xt, At), t = 1, 2.
+The game’s information structure is
+HAlice
+1
+={X1}
+(57)
+HAlice
+2
+={X1, AAlice
+1
+, X2, Z2}
+(58)
+HBob
+1
+=∅
+(59)
+HBob
+2
+={Z2}
+(60)
+where HAlice
+t
+, HBob
+t
+, t = 1, 2, describe the information available to Alice and
+Bob, respectively, at stages 1 and 2.
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+23
+This example has the same dynamics and utility functions as Example 3
+in Tang et al (2022), but Bob doesn’t observe Alice’s action as in (Tang et al,
+2022, Example 3).
+6.2 Sequential decomposition
+Since Alice perfectly observes the state at both times, i.e., Y Alice
+1
+= X1 and
+Y Alice
+2
+= X2, and Bob doesn’t have private information, SAlice
+1
+= X1, SBob
+1
+= ∅
+are sufficient private information for Alice and Bob at stage t = 1, respectively,
+and SAlice
+2
+= X2, SBob
+2
+= ∅ are sufficient private information for Alice and
+Bob, respectively, at stage t = 2 according to Definition 1.
+Suppose σ = (σ1, σ2) = (σAlice
+1
+, σBob
+2
+) is a SIB strategy and ψσ is the
+corresponding update rule. Here σ is an equilibrium strategy candidate which
+serves as the strategy prediction for Alice and Bob. Note that Πψσ,Alice
+1
+(x1) =
+µ0(x1) and Πψσ,Bob
+1
+(x1) = µ0(x1) for all x1 ∈ X1.
+To get a BNE using the sequential decomposition of Theorem 2, we first
+consider the stage-game G2(0, πψσ
+2 ) at time 2. Since Bob is the only agent who
+acts at time 2 and SBob
+2
+= ∅, any BNE σ2 of G2(0, πψσ
+2 ) must satisfy
+ˆσBob
+2
+= arg max
+˜σBob
+2
+E˜σBob
+2
+,ψσ[uBob
+2
+(X2, A2)]
+= arg max
+˜σBob
+2
+�
+− 2 P˜σBob
+2
+,ψσ
+(X2 = ABob
+2
+= 1) − P˜σBob
+2
+,ψσ(X2 = ABob
+2
+= −1)
+�
+= arg max
+˜σBob
+2
+�
+− 2πψσ,Bob
+2
+(1)˜σψσ,Bob
+2
+(1 | πψσ
+2 )
+− (1 − πψσ,Bob
+2
+(1))(1 − ˜σψσ,Bob
+2
+(1 | πψσ
+2 ))
+�
+(61)
+From (61) we conclude that one of the equilibrium SIB strategies is given by
+σBob
+2
+(πψσ
+2 ) = 1, if πψσ,Bob
+2
+(1) ≤ 1/3,
+σBob
+2
+(πψσ
+2 ) = 0, if πψσ,Bob
+2
+(1) > 1/3,
+or equivalently
+σBob
+2
+(πψσ
+2 ) = 1(πψσ,Bob
+2
+(1) ≤ 1/3)
+(62)
+Note that σBob
+2
+(πψσ
+2 ) can take any value in [0, 1] if πψσ,Bob
+2
+(1) = 1/3 and σ2 is
+still a BNE of the stage-game.
+Alice’s sufficient private information at time 2 is SAlice
+2
+= X2. With the
+stage-game equilibrium SIB strategy σBob
+2
+(π2) given by (62), the value function
+for Alice at t = 2 is then given, according to (48), by
+V Alice
+2
+(πψσ
+2 , x2) = Eσ2,ψσ[uAlice
+2
+(X2, A2) | x2]
+
+Springer Nature 2021 LATEX template
+24
+Dynamic Games with Asymmetric Information and Hidden Actions
+=
+�
+21(πψσ,Bob
+2
+(1) ≤ 1/3)
+if x2 = 1
+1 − 1(πψσ,Bob
+2
+(1) ≤ 1/3) if x2 = −1
+(63)
+Given the above value functions at time t = 2, we now consider the stage-
+game G1(V2, πψσ
+1 ) at time t = 1. The utility for the stage-game for Alice is
+given as follows.
+U Alice
+G1(V2,πψσ
+1
+) = uAlice
+1
+(X1, A1) + V Alice
+2
+(ψσ
+2 (π1, Z), X2)
+(64)
+If Alice uses the SIB strategy ˜σAlice
+1
+, the expected utility of the stage-game
+can be calculated for X1 = −1 and X1 = 1, according to (46), by
+E˜σAlice
+1
+,ψσ[U Alice
+G1(V2,πψσ
+1
+) | X1 = −1]
+=c˜σAlice
+1
+(1 | −1) + E˜σAlice
+1
+,ψσ[V A
+2 (ψσ
+2 (πψσ
+1 , X1W1), X1AAlice
+1
+) | X1 = −1]
+=(1 + c)(1 − ˜α1) + (3˜α1 − 1)((1 − p)1(q−1 ≤ 1/3) + p1(q1 ≤ 1/3))
+=:rA
+−1(˜α1, q)
+(65)
+E˜σAlice
+1
+,ψσ[U Alice
+G1(V2,πψσ
+1
+) | X1 = 1]
+=c˜σAlice
+1
+(1 | 1) + E˜σAlice
+1
+,ψσ[V A
+2 (ψσ
+2 (πψσ
+1 , X1W1), X1AAlice
+1
+) | X1 = 1]
+=1 + (c − 1)˜α2 + (3˜α2 − 1)((1 − p)1(q1 ≤ 1/3) + p1(q−1 ≤ 1/3))
+=:rA
+1 (˜α2, q)
+(66)
+where q = (q−1, q1), q−1 = ψσ,Bob
+2
+(πψσ
+1 , −1)(1) and q1 = ψσ,Bob
+2
+(πψσ
+1 , 1)(1) are
+the CIB beliefs πψσ,Bob
+2
+(1) of {X2 = 1} when Z = −1 and Z = 1, respectively,
+and ˜α = (˜α1, ˜α2), ˜α1 = ˜σAlice
+1
+(−1 | −1), ˜α2 = ˜σAlice
+1
+(1 | 1) represents Alice’s
+SIB strategy ˜σAlice
+1
+.
+Note that from Bayes’ rule in Definition 4, under the SIB strategy σAlice
+1
+,
+represented by α1 = σAlice
+1
+(−1 | −1) and α2 = σAlice
+1
+(1 | 1), we have
+q−1 = ψψσ,Bob
+2
+(πψσ
+1 , −1)(1) = Pα(X2 = 1, Z = −1)
+Pα(Z = −1)
+= α2p + α1(1 − p)
+(67)
+q1 = ψψσ,Bob
+2
+(πψσ
+1 , 1)(1) = Pα(X2 = 1, Z = 1)
+Pα(Z = 1)
+= α2(1 − p) + α1p
+(68)
+Therefore, a SIB strategy ˆσAlice
+1
+, represented by ˆα1 = ˆσAlice
+1
+(−1 | −1) and
+ˆα2 = ˆσAlice
+1
+(1 | 1), is a BNE of the stage-game G1(V2, πψσ
+1 ) at time t = 1 if
+ˆα1 ∈ arg max
+˜α1
+rA
+−1(˜α1, (α2p + α1(1 − p), α2(1 − p) + α1p))
+(69)
+ˆα2 ∈ arg max
+˜α2
+rA
+1 (˜α2, (α2p + α1(1 − p), α2(1 − p) + α1p))
+(70)
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+25
+Consequently, the SIB strategy σAlice
+1
+, represented by α1 = σAlice
+1
+(−1 | −1)
+and α2 = σAlice
+1
+(1 | 1) will satisfy the sequential decomposition equations
+(49)-(50) if
+α1 ∈ arg max
+˜α1
+rA
+−1(˜α1, (α2p + α1(1 − p), α2(1 − p) + α1p))
+(71)
+α2 ∈ arg max
+˜α2
+rA
+1 (˜α2, (α2p + α1(1 − p), α2(1 − p) + α1p))
+(72)
+Remark 10. Note that the functions rA
+−1(˜α1, q) and rA
+1 (˜α2, q) are not contin-
+uous in q. Thus existence of equilibria cannot be established by the standard
+method relying on the continuity of the utility functions, and there may not no
+equilibria in the general case.
+6.3 Existence of SIB-BNE under conditions on the
+instantaneous utility.
+The stage-game G1(V2, πψσ
+1 ) is a normal-form game with a fixed σ1. According
+to Remark 5, a BNE ˆσ of G1(V2, πψσ
+1 ) could be different from σ1 and the
+existence of a regular BNE of G1(V2, πψσ
+1 ) is not sufficient to satisfy (50) at
+time t = 1. In order to apply equilibrium existence results for normal-form
+games to the sequential decomposition at time t = 1, we introduce an agent 0
+who picks the q-belief q = (q−1, q1) so that (50) is satisfied.
+Formally, we construct an augmented stage-game ˆG1 between Alice and
+agent 0. Alice chooses ˜α = (˜α1, ˜α2) and agent 0 chooses ˜q = (˜q−1, ˜q1). Alice’s
+utility is
+rA
+1 (˜α, ˜q) =0.5rA
+−1(˜α1, ˜q) + 0.5rA
+1 (˜α2, ˜q)
+=0.5c(1 − ˜α1 + ˜α2) + 0.5(2 − ˜α1 − ˜α2)
++ 0.5(3(˜α2p + ˜α1(1 − p)) − 1)1(˜q−1 ≤ 1/3)
++ 0.5(3(˜α2(1 − p) + ˜α1p) − 1)1(˜q1 ≤ 1/3).
+(73)
+Agent 0’s utility is
+r0
+1(˜α, ˜q) = −(˜q−1 − ˜α2p − ˜α1(1 − p))2 − (˜q1 − ˜α2(1 − p) − ˜α1p)2.
+(74)
+Both Alice and agent 0 are utility maximizers. The game ˆG1 with utilities
+(74)-(73) is a normal-form game with strategies ˜α = (˜α1, ˜α2) ˜q = (˜q−1, ˜q1).
+Since the utility (74) of agent 0 is a quadratic function, any best response by
+agent 0 must satisfy ˜q−1 = ˜α2p + ˜α1(1 − p), ˜q1 = ˜α2(1 − p) + ˜α1p.
+Note that in the augmented stage-game ˆG1, the utility function rA
+1 (˜α, ˜q)
+is not continuous in ˜q. To show the existence of a Nash equilibrium for ˆG1,
+we proceed to apply existence results for games with discontinuous utilities in
+Barelli and Meneghel (2013).
+
+Springer Nature 2021 LATEX template
+26
+Dynamic Games with Asymmetric Information and Hidden Actions
+Specifically, Proposition 2.4 of Barelli and Meneghel (2013) guarantees the
+existence of a Nash equilibrium for games satisfying the generalized better
+reply secure property. From Definition 2.3 in Barelli and Meneghel (2013), the
+stage game is generalized better reply secure if for any (¯α, ¯q) not an equilibrium,
+at least one of the followings is true
+• We can find an ǫ > 0 and a closed correspondence φ0(˜α, ˜q) such that
+r0
+1(˜α, φ0(˜α, ˜q)) ≥ r0
+1(¯α, ¯q) + ǫ
+(75)
+for all ˜α1 ∈ (¯α1 − ǫ, ¯α1 + ǫ), ˜α2 ∈ (¯α2 − ǫ, ¯α2 + ǫ), ˜q−1 ∈ (¯q−1 − ǫ, ¯q−1 + ǫ),
+˜q1 ∈ (¯q1 − ǫ, ¯q1 + ǫ)
+• We can find an ǫ > 0 and a closed correspondence φA(˜α, ˜q) such that
+rA
+1 (φA(˜α, ˜q), ˜q) ≥ rA
+1 (¯α, ¯q) + ǫ
+(76)
+for all ˜α1 ∈ (¯α1 − ǫ, ¯α2 + ǫ), ˜α2 ∈ (¯α2 − ǫ, ¯α2 + ǫ), ˜q−1 ∈ (¯q−1 − ǫ, ¯q−1 + ǫ),
+˜q1 ∈ (¯q1 − ǫ, ¯q1 + ǫ)
+In Appendix .2, we show that when c > 24 the augmented stage-game
+ˆG1 is generalized better reply secure. Thus, there exists a Nash equilibrium
+of the augmented state-game ˆG1 according to (Barelli and Meneghel, 2013,
+Proposition 2.4).
+Consider any Nash equilibrium (α, q) of ˆG1. Since q is a best response to
+α for agent 0, from agent 0’s utility (74) we have
+q−1 = α2p + α1(1 − p)
+(77)
+q1 = α2(1 − p) + α1p
+(78)
+Furthermore, since α is a best response to q for Alice in ˆG1,
+α ∈ arg max
+˜α
+�
+0.5rA
+−1(˜α1, q) + 0.5rA
+1 (˜α2, q)
+�
+= arg max
+˜α
+�
+0.5rA
+−1(˜α1, (α2p + α1(1 − p), α2(1 − p) + α1p))
++ 0.5rA
+1 (˜α2, (α2p + α1(1 − p), α2(1 − p) + α1p))
+�
+=
+�
+arg max
+˜α1
+rA
+−1(˜α1, (α2p + α1(1 − p), α2(1 − p) + α1p)),
+arg max
+˜α2
+rA
+1 (˜α2, (α2p + α1(1 − p), α2(1 − p) + α1p))
+�
+(79)
+Therefore, (71)-(72) hold for α, and consequently the sequential decomposi-
+tion requirement (49)-(50) is satisfied at t = 1 by the SIB strategy σAlice
+1
+represented by α, and we establish the existence of a SIB equilibrium based
+on Theorem 2.
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+27
+7 The case with no common observations
+We consider the model of Section 2 but we assume that the agents have no
+common observations, that is,
+Zt = ∅
+∀t ∈ T .
+(80)
+The system’s dynamics, the agents’ private observations, the functional form
+of the agents’ strategies, their utilities, and the equilibrium concept (BNE)
+remain the same as in Section 2.
+Even though the agents have no common observations in this special case,
+we can still define SIB strategies by Definition 3, and construct the consistent
+CIB belief system according to Definition 4 with Zt = ∅ ∀t ∈ T .
+Since there is no common observations, for any realization we always have
+�
+ˆxt+1,ˆst+1
+F i
+t (ˆxt+1, ˆst+1, zt+1)(πψσ
+t
+; σ−i
+t )
+=
+�
+ˆxt+1,ˆst+1
+F i
+t (ˆxt+1, ˆst+1)(πψσ
+t
+; σ−i
+t ) = 1 > 0
+(81)
+Therefore, case (ii) in Definition 4 would never happen, and (20) can be
+simplified to
+πψσ,i
+t+1 (xt+1, st+1)
+=
+F i
+t (xt+1, st+1)(πψσ
+t
+; σ−i
+t )
+�
+ˆxt+1,ˆst+1 F i
+t (ˆxt+1, ˆst+1)(πψσ
+t
+; σ−i
+t )
+=F i
+t (xt+1, st+1)(πψσ
+t
+; σ−i
+t )
+=
+�
+yt+1,xt,st,at
+�
+P{yt+1, xt+1 | xt, at}
+��
+j
+1{sj
+t+1 = φj
+t+1(sj
+t, yj
+t+1, aj
+t)}
+�
+
+ 1
+|Ai
+t|
+�
+j̸=i
+σj
+t (aj
+t)(πψ
+t , sj
+t)
+
+ πψ,i
+t
+(xt, st)
+�
+.
+(82)
+Based on (82) we can write
+Πψσ,i
+t+1 = ψσ,i
+t+1(Πψσ
+t )
+∀i ∈ N,
+(83)
+Πψσ
+t+1 = ψσ
+t+1(Πψσ
+t ).
+(84)
+In other words, given a SIB strategy σ, the update rule ψσ are deterministic
+functions given by (84), and the corresponding consistent CIB belief system
+Πψσ
+t , t ∈ T , evolves in a deterministic manner. Furthermore, since case (ii)
+in Definition 4 never happens without common observations, the update rule
+
+Springer Nature 2021 LATEX template
+28
+Dynamic Games with Asymmetric Information and Hidden Actions
+ψσ,i
+t+1 given by (82) becomes exactly the Bayes rule. As a result, the CIB belief
+Πψσ,i
+t
+becomes a regular PMF given by
+Πψσ,i
+t
+(xt, st) = P˜gi,σ−i(xt, st)
+∀i ∈ N
+(85)
+where ˜gi denotes the uniform strategy (i.e., the strategy that chooses every
+action ai
+t ∈ Ai
+t with equal probability for all t ∈ T ).
+Remark 11. If the N agents have identical utilities, i.e. we have a dynamic
+team problem, then Πψσ
+t , t ∈ T is similar to the common knowledge that
+appears in Witsenhausen (1973) where a dynamic team is analyzed. The com-
+mon knowledge in Witsenhausen (1973) is a sequence (over time) of PMFs on
+the system’s history Ht, t ∈ T . These PMFs evolve in a deterministic manner,
+similar to (82) for Πψσ
+t , t ∈ T , in the model of this section.
+For this special case with no common observations, Theorem 2 becomes
+Corollary 1. Consider a SIB strategy profile σ = {σt, t ∈ T } and the corre-
+sponding update rule ψσ = {ψσ
+t , t ∈ T } defined by (83)-(84) for the model of
+this section. Define
+V i
+T +1(·, ·) = 0 for all i
+(86)
+V i
+t (πψσ
+t , si
+t) = Eσt,ψσ[U i
+Gt(Vt+1,πψσ
+t ) | si
+t]
+(87)
+where U i
+Gt(Vt+1,πψσ
+t
+)
+=
+ui
+t(Xt, At) + V i
+t+1(ψσ
+t+1(πψσ
+t
+), Si
+t+1), and in the
+conditional expectation Eσt,ψσ[·], the distribution of (Xt, St) conditioned
+on Si
+t is given by πψσ,i
+t
+(xt, s−i
+t ), Ai
+t, i
+∈
+N, are generated by σi
+t(ai
+t
+|
+si
+t, πψσ
+t
+), Si
+t+1 conditioned on (Xt, St, At) follows the conditional probability
+�
+xt+1,s−i
+t+1 P(xt+1, st+1 | xt, st, at) given by
+P(xt+1, st+1 | xt, st, at)
+=
+�
+yt+1
+P{xt+1 | xt, at}P{yt+1 | xt+1, at}
+��
+j
+1{sj
+t+1 = φj
+t+1(sj
+t, yj
+t+1, aj
+t)}
+�
+.
+(88)
+If for all t ∈ T , there is a SIB strategy profile ˆσt such that ˆσt is a BNE of the
+stage-game Gt(Vt+1, πψσ
+t ), that is,
+Eˆσi
+t,ˆσ−i
+t
+,ψσ[U i
+Gt(Vt+1,πψσ
+t
+) | si
+t] = max
+˜σi
+t∈Λi
+t
+E˜σi
+t,ˆσ−i
+t
+,ψσ[U i
+Gt(Vt+1,πψσ
+t
+) | si
+t]
+(89)
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+29
+for all i ��� N, and
+ˆσt = σt,
+(90)
+then the SIB strategy profile σ is a SIB-BNE of the dynamic game without
+common observations defined in this section.
+Remark 12. The SIB-BNE strategy profiles {σt, t ∈ T } determined by sequen-
+tial decomposition in Corollary 1, along with the beliefs {Πψσ
+t , t ∈ T } are also
+Perfect Bayesian Equilibria (PBE) Fudenberg and Tirole (1991). This is true
+because {σt, t ∈ T } satisfy sequential rationality (Eq. (89)) and consistency
+holds because the beliefs {Πψσ
+t , t ∈ T } are always updated by Bayes rule.
+8 Conclusion
+We considered stochastic dynamic games where the underlying system is
+dynamic, the strategic agents’ actions are hidden (not observable) and their
+information is asymmetric. We presented an approach for the computation of
+BNE strategy profiles that are based on a compressed version of the agents’
+information and can be determined sequentially in time moving backwards, if
+each step of this backward procedure has a solution. The approach highlights:
+(i) the importance of common information/common knowledge in identifying
+BNE strategy profiles that can be sequentially computed; (ii) the difference
+between common information that is sufficient for decision-making purposes
+in games and common information that is sufficient for decision-making pur-
+poses in teams. The difference is due to the fact that agents have an incentive
+to deviate from their predicted strategies in games whereas they don’t have
+such an incentive in teams. As a consqence of this incentive, at each time
+instant each agent has his own view/belief of the game’s status based on the
+common information, but all these different views/beliefs are common knowl-
+edge among all agents. As a result the CIB belief system is described by the
+sequence Πψ
+1:T specified by Definition 2.
+Our investigation focused on determining SIB-BNE strategy profiles for
+the games under consideration. We note that the SIB-BNE strategy profiles
+determined by our methodology are also Perfect Bayesian Equilibrium (PBE)
+strategy profiles when the agents have no common observations (i.e., for the
+model of Section 7), but this is not true when the agents have common obser-
+vations (the general model of Section 2). Determining PBE strategy profiles for
+the general model of Section 2 is an interesting problem worthy of investigation.
+.1 Sufficient Information
+We compare conditions (i)-(iii) of Definition 1 to the conditions of Definition
+2 in Tavafoghi et al (2022); for ease of readability, we include the definition
+from Tavafoghi et al (2022) below.
+
+Springer Nature 2021 LATEX template
+30
+Dynamic Games with Asymmetric Information and Hidden Actions
+Definition 7 (Sufficient private information Tavafoghi et al (2022)). We say
+Si
+t = ζi
+t(P i
+t , Ct; g1:t−1), i ∈ N, t ∈ T , is sufficient private information for the
+agents if,
+(i) it can be updated recursively as
+Si
+t = φi
+t(Si
+t−1, Hi
+t\Hi
+t−1; g1:t−1) for t ∈ T \{1},
+(91)
+(ii) for any strategy profile g and for all realizations {ct, pt, pt+1, zt+1, at} ∈
+Ct × Pt × Pt+1 × Zt+1 of positive probability,
+Pg1:t {st+1,zt+1 | pt,ct,at}=Pg1:t {st+1,zt+1 | st,ct,at},
+(92)
+where s1:N
+τ
+= ζ1:N
+τ
+(p1:N
+τ
+, cτ; g1:τ−1) for τ ∈ T ;
+(iii) for every strategy profile ˜g of the form ˜g:={˜gi
+t : Si
+t × Ct → ∆(Ai
+t), i∈N,t∈
+T } and at∈At, t∈T ;
+E˜g1:t−1�
+ui
+t(Xt,At) | ct,pi
+t,at
+�
+=E˜g1:t−1�
+ui
+t(Xt,At) | ct,si
+t,at
+�
+,
+(93)
+for all realizations {ct,pi
+t} ∈ Ct × Pi
+t of positive probability where s1:N
+τ
+=
+ζ1:N
+τ
+(p1:N
+τ
+,cτ; ˜g1:τ−1) for τ ∈ T ;
+(iv) given an arbitrary strategy profile ˜g of the form ˜g := {˜gi
+t : Si
+t × Ct →
+∆(Ai
+t), i∈N, t∈T }, i∈N, and t∈T ,
+P˜g1:t−1�
+s−i
+t
+| pi
+t,ct
+�
+=P˜g1:t−1�
+s−i
+t
+| si
+t,ct
+�
+,
+(94)
+for all realizations {ct,pi
+t} ∈ Ct ×Pi
+t of positive probability where s1:N
+τ
+=
+ζ1:N
+τ
+(p1:N
+τ
+,cτ; ˜g1:τ−1) for τ ∈ T .
+Condition (i) of Definition 1 appears in the definition of Si
+t in Definition 7,
+and condition (ii) of Definition 1 on recursive update is the same as condition
+(i) in Definition 7. Condition (iii) of Definition 1 directly leads to (iii) and (iv)
+of Definition 7; the utility ui
+t(Xt, At) in condition (iii) and the random variable
+s−i
+t
+in condition (iv) of Definition 7 are functions of (xt, st) whose distribution
+conditioned on (pi
+t, ct) is the same as conditioned on (si
+t, ct) under condition
+(iii) of Definition 1.
+However, condition (ii) of Definition 7 may not hold for sufficient private
+information satisfying Definition 1. Consider the following example. Suppose
+X1 = Y 1
+1 XOR Y 2
+1 , and Y 1
+1 , Y 2
+1 takes values in {0, 1} with equal probability.
+Z1 = ∅ and Z2 = X1. Then S1
+1 = S2
+1 = ∅ satisfies Definition 1 because
+P(x1, s−i
+1
+| pi
+1, c1) = P(x1 | yi
+1) = 0.5 = P(x1, s−i
+1
+| si
+1, c1). However, they don’t
+satisfy condition (ii) of Definition 7 because P(z2 | p1, c1, a1) = P(x1 | y1
+1, y2
+1) =
+1(x1 = y1
+1 XOR y2
+1) ̸= P(z2 | s1, c1, a1) = P(x1) = 0.5.
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+31
+.2 Proof of the generalized better reply secure property
+for the augmented stage-game
+We show that when c > 24 the augmented stage-game ˆG1 in Section 6 is
+generalized better reply secure. For that matter, we set β∗(q) = 1(q ≤ 1/3)
+and consider the following five cases.
+Case (i) r0
+1(¯α, ¯q) ̸= 0. In this case Bayes’ rule doesn’t hold at (¯α, ¯q). We focus
+on agent 0 and select the belief to satisfy Bayes’ rule as follows:
+φ0(˜α, ˜q) = (˜α2p + ˜α1(1 − p), ˜α2(1 − p) + ˜α1p)
+(95)
+Then this φ0 is a closed correspondence. From this construction of φ0,
+we can pick ǫ > 0 such that
+r0
+1(˜α, φ0(˜α, ˜q)) = 0 > r0
+1(¯α, ¯q) + ǫ
+Case (ii) r0
+1(¯α, ¯q) = 0, and ¯π−1 ̸= 1/3 and ¯π1 ̸= 1/3.
+Since β∗(q) = 1 if q < 1/3, β∗(q) = 0 if q > 1/3, β∗(·) is continuous
+at points where q ̸= 1/3. Hence, we can find ǫ > 0 s.t. β∗(˜q−1) =
+β∗(¯q−1) for all ˜q−1 ∈ (¯q−1 − ǫ, ¯q−1 + ǫ), and β∗(˜q1) = β∗(¯q1) for all
+˜q1 ∈ (¯q1 − ǫ, ¯q1 + ǫ). In this region we have
+rA
+1 (α, ˜q) = rA
+1 (α, ¯q)
+(96)
+for all α. Let
+φA(˜α, ˜q) = arg max
+α
+rA
+1 (α, ˜q)
+(97)
+Because rA
+1 (·) is continuous in the region under consideration, φA(·)
+has a closed graph from Berge’s maximum theorem. Note that for all
+˜q1 ∈ (¯q1 − ǫ, ¯q1 + ǫ), ˜q1 ∈ (¯q1 − ǫ, ¯q1 + ǫ)
+rA
+1 (φA(˜α, ˜q), ˜q) = max
+α
+rA
+1 (α, ˜q) = max
+α
+rA
+1 (α, ¯q)
+(98)
+If maxα rA
+1 (α, ¯q) > rA
+1 (¯α, ¯q) we can find ǫ > 0 such that for ˜q1 ∈
+(¯q1 − ǫ, ¯q1 + ǫ), ˜q1 ∈ (¯q1 − ǫ, ¯q1 + ǫ), rA
+1 (φA(˜α, ˜q), ˜q) = maxα rA
+1 (α, ˜q) ≥
+rA
+1 (¯α, ¯q) + ǫ.
+If maxα rA
+1 (α, ¯q) = rA
+1 (¯α, ¯q), then Alice has no profitable deviation.
+Furthermore, since r0
+1(¯α, ¯q) = 0, agent 0 has no profitable deviation.
+Consequently, (¯α, ¯q) is an equilibrium if maxαrA
+1 (α, ¯q) = rA
+1 (¯α, ¯q).
+Case (iii) r0
+1(¯α, ¯q) = 0, ¯π−1 = 1/3 and ¯π1 ̸= 1/3.
+Note that ¯q−1 = 0.8¯α1+0.2¯α2 = 1/3 and β∗(¯q−1) = 1/3. Since ¯π1 ̸=
+1/3, we can find ǫ > 0 s.t. β∗(˜q1) = β∗(¯q1) for all ˜q1 ∈ (¯q1 − ǫ, ¯q1 + ǫ).
+
+Springer Nature 2021 LATEX template
+32
+Dynamic Games with Asymmetric Information and Hidden Actions
+Therefore,
+rA
+1 (¯α, ¯q) = 0.5c(1 − ¯α1 + ¯α2) + 0.5(2 − ¯α1 − ¯α2) + 0.5(3¯q1 − 1)β∗(¯q1)
+(99)
+Pick for Alice
+φA(˜α, ˜q) = (0, 1)
+(100)
+for all ˜αi ∈ (¯αi −ǫ, ¯αi+ǫ), i = 1, 2, ˜qi ∈ (¯qi −ǫ, ¯qi+ǫ), i = −1, 1. We get
+rA
+1 (φA(˜α, ˜q), ˜q) =c + 0.5 + 0.5(0.6 − 1)β∗(˜q−1) + 0.5(2.4 − 1)β∗(˜q−1)
+=c + 0.5 − 0.2β∗(˜q−1) + 0.7β∗(¯q−1)
+(101)
+and
+rA
+1 (φA(˜α, ˜q), ˜q) − rA
+1 (¯α, ¯q) − ǫ
+=0.5c(1 + ¯α1 − ¯α2) − 0.5(1 + ¯α1 + ¯α2)
+− 0.2β∗(˜q−1) + 0.5(2.4 − 3¯q1)β∗(¯q−1) − ǫ
+≥0.5c(1 + ¯α1 − ¯α2) − 0.5 ∗ 3 − 0.2 − 0.5 ∗ 0.6 − ǫ
+(102)
+When ¯q−1 = 1/3, then 0.8¯α1 + 0.2¯α2 = 1/3 ⇒ ¯α1 = 5/12 − 3/12¯α2.
+Therefore,
+1 + ¯α1 − ¯α2 = 17/12 − 15/12¯α2 ≥ 1/6
+(103)
+where the minimum is at ¯α1 = 1/6 and ¯α2 = 1.
+When c > 24, then
+0.5c(1 + ¯α1 − ¯α2) ≥ c/12 > 2
+(104)
+and rA
+1 (φA(˜α, ˜q), ˜q) − rA
+1 (¯α, ¯q) − ǫ > 0.
+Case (iv) r0
+1(¯α, ¯q) = 0, and ¯π1 = 1/3 and ¯π−1 ̸= 1/3.
+This case is similar to case (iii). Since ¯π−1 ̸= 1/3, we can find ǫ > 0
+s.t. β∗(˜q−1) = β∗(¯q−1) for all ˜q−1 ∈ (¯q−1 − ǫ, ¯q−1 + ǫ). Furthermore,
+rA
+1 (¯α, ¯q)
+=0.5c(1 − ¯α1 + ¯α2) + 0.5(2 − ¯α1 − ¯α2) + 0.5(3¯q−1 − 1)β∗(¯q−1)
+(105)
+Pick for Alice the closed correspondence (as in case (iii))
+φA(˜α, ˜q) = (0, 1)
+(106)
+
+Springer Nature 2021 LATEX template
+Dynamic Games with Asymmetric Information and Hidden Actions
+33
+for all ˜αi ∈ (¯αi − ǫ, ¯αi + ǫ), i = 1, 2, ˜qi ∈ (¯qi − ǫ, ¯qi + ǫ), i = −1, 1. Then
+rA
+1 (φA(˜α, ˜q), ˜q)
+=c + 0.5 − 0.2β∗(¯q−1) + 0.7β∗(˜q−1)
+(107)
+and
+rA
+1 (φA(˜α, ˜q), ˜q) − rA
+1 (¯α, ¯q) − ǫ
+=0.5c(1 + ¯α1 − ¯α2) − 0.5(1 + ¯α1 + ¯α2)
++ 0.5(0.6 − 3¯q−1)β∗(¯q−1) + 0.7β∗(˜q−1) − ǫ
+≥0.5c(1 + ¯α1 − ¯α2) − 0.5 ∗ 3 − 0.5 ∗ 2.4 − ǫ
+(108)
+When ¯q1 = 1/3, 0.2¯α1+0.8¯α2 = 1/3 ⇒ ¯α2 = 5/12−3/12¯α1. Therefore,
+1 + ¯α1 − ¯α2 = 7/12 + 15/12¯α1 ≥ 7/12.
+(109)
+When c > 24, then
+0.5c(1 + ¯α1 − ¯α2) ≥ 7/24c > 2.7
+(110)
+and rA
+1 (φA(˜α, ˜q), ˜q) − rA
+1 (¯α, ¯q) − ǫ > 0.
+Case (v) r0
+1(¯α, ¯q) = 0, and ¯π1 = 1/3 and ¯π−1 = 1/3.
+We have
+rA
+1 (¯α, ¯q) = 0.5c(1 − ¯α1 + ¯α2) + 0.5(2 − ¯α1 − ¯α2)
+(111)
+Pick for Alice the closed correspondence (as in cases (iii) and (iv))
+φA(˜α, ˜q) = (0, 1)
+(112)
+for all ˜αi ∈ (¯αi − ǫ, ¯αi + ǫ), i = 1, 2, ˜qi ∈ (¯qi − ǫ, ¯qi + ǫ), i = −1, 1. Then
+rA
+1 (φA(˜α, ˜q), ˜q) − rA
+1 (¯α, ¯q) − ǫ
+=0.5c(1 + ¯α1 − ¯α2) − 0.5(1 + ¯α1 + ¯α2) − 0.2β∗(˜q−1) + 0.7β∗(˜q−1) − ǫ
+≥0.5c(1 + ¯α1 − ¯α2) − 0.5 ∗ 3 − 0.2 − ǫ
+(113)
+Then we have rA
+1 (φA(˜α, ˜q), ˜q) − rA
+1 (¯α, ¯q) − ǫ > 0 following the steps in
+(iv).
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+

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+A quantum pricing-based column generation framework for hard combinatorial problems
+Wesley da Silva Coelho,1, ∗ Lo¨ıc Henriet,1 and Louis-Paul Henry1
+1PASQAL SAS, 7 rue L´eonard de Vinci, 91300 Massy, France
+(Dated: January 9, 2023)
+In this work, we present a complete hybrid classical-quantum algorithm involving a quantum
+sampler based on neutral atom platforms. This approach is inspired by classical column generation
+frameworks developed in the field of Operations Research and shows how quantum procedures can
+assist classical solvers in addressing hard combinatorial problems. We benchmark our method on the
+Minimum Vertex Coloring problem and show that the proposed hybrid quantum-classical column
+generation algorithm can yield good solutions in relatively few iterations. We compare our results
+with state-of-the-art classical and quantum approaches.
+INTRODUCTION
+Combinatorial optimization is at the heart of many real-
+world problems. It consists in finding the “best” out of a
+finite, but prohibitively large, set of options. Column gener-
+ation [1] is an iterative method that was developed to solve
+this kind of difficult mathematical problems, such as linear
+formulations where the problem may be too large to consider
+all options explicitly. In this method, variables are associated
+with each option. The algorithm starts by solving the con-
+sidered problem with a limited set of variables (or options),
+known as restricted master problem (RMP), and then itera-
+tively adds variables to improve the objective function. The
+generation of new variables is done by an algorithm specifi-
+cally tailored to this task: during each iteration, the related
+new sub-problem to be solved, usually referred to as pric-
+ing sub-problem (PSP), relies on the duality theory [2] to
+provide new variables, if there exists any, only if they can
+improve the current solution of the restricted master prob-
+lem. The iterative process stops when new variables cease to
+improve the objective function, which is proven mathemati-
+cally. However, solving the pricing sub-problems usually rep-
+resents the bottleneck of the column generation approach as
+it comes to solving several simpler, but still hard, optimiza-
+tion problems. Hence, designing an efficient way to solve the
+pricing sub-problems is the most important step to ensure
+high-quality solutions to the RMP while minimizing time
+and resource consumption.
+During the past years, both academic and industrial com-
+munities have been putting a lot of effort into designing
+quantum hardware and algorithms that could provide a real
+advantage over classical computers. As pointed out in [3],
+this advantage can take the form of more accurate results,
+a faster convergence, or even a lower energy consumption.
+Such quantum algorithms can be used along with state-of-
+the-art classical solutions, such as the column generation al-
+gorithm, to create powerful hybrid classical-quantum frame-
+works. A wide spectrum of quantum computing platforms
+is currently being developed, using different kinds of two-
+level systems as qubits, including Josephson junctions [4, 5],
+trapped ions [6, 7], photons [8], or neutral atoms [9, 10].
+Each of these allows for different quantum processing unit
+(QPUs) architectures, with their own advantages and limi-
+tations when it comes to the connectivity of the qubits, or
+the types of operations that are easily implemented. A good
+knowledge of these platforms allows for the development of
+hardware-efficient approaches, designed specifically for each
+∗ wesley.coelho@pasqal.com
+FIG. 1: Workflow of the hybrid classical-quantum column
+generation approach. First, a minimal sub-set x of variables
+is generated in such a way that it ensures a feasible solu-
+tion for the Reduced Master Problem (e.g., with only the
+singletons of the graph). The RMP is then solved by a clas-
+sical solver. The next steps are related to the pricing sub-
+problems, which are solved by considering the dual values
+from the solved RMP in order to find more variables that
+can potentially improve the current solution of the RMP.
+If such variables exist, then RMP is updated with the new
+variables and is solved again. The search for new variables is
+done by a quantum sampler specifically tailored to consider
+different inputs related to each pricing iteration. These last
+steps are repeated until no column is generated by the PSP.
+of them. In particular, this allows for identifying the classi-
+cal bottlenecks that are best suited for being replaced by a
+quantum approach.
+In this work, we propose a complete hybrid classical-
+quantum column generation framework whose pricing sub-
+problems can be efficiently solved on neutral atom-based
+QPUs. Unlike other hybrid approaches like QAOA, the core
+part of the resolution is here carried out by a classical solver,
+and the quantum processing unit is used as a sampler to
+restrict the search space. Requiring only |V | qubits for a
+given graph G
+= (V, E), the related pricing sub-problems
+are then solved by a neutral atom-based sampler specifically
+tailored to improve the current solution of the associated
+master problem. Compared to classical and quantum greedy
+approaches, we show numerically that the proposed hybrid
+column generation can improve significantly the quality of
+the solutions while reducing the number of iterations on the
+quantum device. Finally, by taking advantage of some quan-
+arXiv:2301.02637v1  [quant-ph]  6 Jan 2023
+
+Pre-Processing
+Solving RMP
+Post-Processing
+Master Problem
+(Re)Build RMP
+Classical
+Graph Weighting
+Solver
+Feasible Solutions
+min cty
+s.t.
+Ay ≤B
+Pre-Processing
+Sampling
+Sampled Solutions
+Variable Set Update
+Register Design
+BS Count Cost
+Quantum Sampler
+00000
+0
+Pricing
+000111
+100
+30
+QPU
+01010
+6
+0.5
+Pulse Shaping
+01011
+250
+40
+00101
+3
+1
+11111
+0
+-102
+tum features (e.g., state superposition), we find that the hy-
+brid column generation method returns the (near-)optimal
+solution faster than the classical one (i.e., where no QPU is
+involved). Fig. 1 summarizes our proposed approach.
+This paper is structured as follows : We first introduce the
+main aspects of the graph theory and the related combina-
+torial problems in Section I. After reviewing related works
+in Section II, we give a brief introduction to neutral atom-
+based quantum computing the Section III. We dedicate Sec-
+tion IV to introduce the main idea of the column generation
+approach. In Section V, we present in-depth our proposed
+hybrid classical-quantum approach to solving the Minimum
+Vertex Coloring Problem (MVCP), while the results of our
+numerical experiments are discussed in Section VI.
+I.
+BACKGROUND
+Combinatorial problems [11] have been extensively stud-
+ied by both academic and industrial communities, and have
+a vast range of applications in real-world systems.
+Those
+problems can naturally be defined on graphs, which are data
+structures composed of a set of elements called vertices (also
+known as nodes) that can potentially be connected. These
+connections are called edges and might potentially encode
+different information, such as the importance of such connec-
+tions (as weights) or the distance between their endpoints.
+Similarly, different labels and weights can also be associ-
+ated with vertices in order to differentiate them. Formally, a
+graph G = (V, E) is composed of a set of vertices V and edges
+E ∈ V2 representing the existence of a connection between
+vertices u and v from V.
+Several real-world optimization problems, from a vast
+spectrum of fields, can be mapped to graph problems. For in-
+stance, graphs can be used to encode social experiments [12],
+telecommunication networks [13], and physical systems [14].
+The related optimization problems typically consist in se-
+lecting a subset of vertices and/or edges optimally satisfying
+certain rules. This kind of discrete optimization problem is
+highly relevant for Quantum Computing (QC), particularly
+in the case of Noisy Intermediate-Scale Quantum (NISQ)-era
+platforms [15, 16]. In that case, results are typically obtained
+via repeated measurements of the final state of the system.
+The solutions are then inferred by the selection of the best
+sampled state through computationally cheap classical post-
+processing.
+In the case of neutral atom QPUs, the spatial arrange-
+ment of qubits can be made such that the Ising Hamiltonian
+describing the interactions in the system is closely related to
+a given cost function to be minimized. This is what makes
+this platform notably well suited to solving graph combina-
+torial problems [17–20]. As the state of the computational
+basis in which the qubits are measured has a direct corre-
+spondence to the solution to the graph problem, this type of
+QC is particularly robust to noise (noise can even be an ad-
+vantage [21]). For instance, Maximum Independent Set [22]
+and Maximum Cut [3] problems can be efficiently solved by
+approaching the ground state of the quantum system with
+adiabatic annealing [18, 23] or similar methods. In the fol-
+lowing, we formally defined some graph problems and show
+how they can be solved by quantum-based approaches.
+A.
+Combinatorial problems
+In the following, we present the Maximum Independent
+set, which is a fundamental part of the proof of concept of
+our hybrid approach. Then, we formally define the Vertex
+coloring problem and present the related mathematical for-
+mulation.
+1.
+Maximum Independent Set problem
+An independent set in a graph G = (V, E) is a subset
+of vertices ˜V ⊂ V such that no pair of elements from ˜V
+is connected by an edge.
+The independent sets of G can
+formally be defined as follows:
+ISG =
+�
+˜V ⊂ V
+�� ˜V2 ∩ E = ∅
+�
+(1)
+where ˜V2 are all the possible edges connecting the vertices in
+˜V. Therefore, the Maximum Independent Set (MIS) is the
+largest set of ISG:
+MIS(G) = argmax
+˜V∈ISG
+|˜V|.
+(2)
+The MIS problems can be alternatively described in
+terms of their quadratic unconstrained binary optimization
+(QUBO) formulations. Consider a graph G = (V, E). Let
+xu be a binary variable associated with each vertex u ∈ V,
+and that holds 1 if vertex u is selected to be in the in-
+dependent set, and 0 otherwise. Hence, the binary vector
+x = {x1, . . . , x|V|} can be put in one-to-one correspondence
+with partitions ˜V of the vertex set V via the identification:
+˜V(x) =
+�
+u ∈ V
+�� xu = 1
+�
+(3)
+The solution to the maximum (weighted) independent set
+problem is then given by:
+MIS(G) = argmin
+x∈{0,1}|V|
+�
+�−
+�
+u∈V
+wuxu + α
+�
+{u,v}∈E
+xuxv
+�
+�
+(4)
+where the parameter wu represents the weight associated
+to each vertex u ∈ V, while α > 0 is an arbitrary coefficient
+to penalize unfeasible solutions. Note that, on unweighted
+graphs, the penalty coefficient α as well as all w parameters
+are set to 1.
+2.
+Minimum Vertex Coloring problem
+The Vertex Coloring problem has several applications in
+real-world optimization problems such as network design [24]
+and task scheduling [25].
+A vertex coloring is an assign-
+ment of colors (or labels) to each vertex of a graph such that
+any two identically colored vertices are not connected by an
+edge. The Minimum Vertex Coloring problem consists then
+in finding a feasible coloring while minimizing the number
+of colors (or labels) assigned; the minimum number of col-
+ors used to color all vertices of a given graph G is called its
+chromatic number, hereafter denoted X(G). The Minimum
+vertex coloring can be formally defined as follows:
+
+3
+(a) Trivial coloring.
+(b) Optimal coloring.
+FIG. 2: Two coloring solutions to the same graph with 5
+vertices, 6 edges, and X(G) = 3. The set C has 5 available
+colors: green, brown, orange, purple, and yellow.
+Definition 1 Let G
+=
+(V, E) be a graph with a set
+V
+=
+{u1, .., u|V|} of vertices and a set E ⊂ V2 of edges.
+Also, let C be a set of available colors. The Minimum Ver-
+tex Coloring Problem consists in coloring each vertex of G
+with exactly one color from C in a such way that the num-
+ber of used colors is minimized while ensuring that no two
+adjacent vertices have the same color.
+Figure 2 shows two possible coloring solutions for the same
+graph. By applying a trivial coloring (see Fig. 2a), each color
+is mapped to exactly one vertex; this simple approach always
+gives a feasible solution to the problem. As shown in Fig. 2b,
+however, the related chromatic number (i.e., the optimal so-
+lution) can be reduced to 3. It is worthwhile to notice that,
+given a feasible solution for the Vertex Coloring problem,
+any sub-set of vertices colored with the same color is also
+an independent set. Hence, finding the minimum sub-set of
+independent sets that cover all vertices of a given graph G is
+equivalent to solving the MVCP in the same graph. Finding
+the chromatic number of a graph, however, is one of Karp’s
+21 NP-complete problems [26].
+B.
+An extended formulation for the Minimum Vertex
+Coloring problem
+We now present an extended formulation1 for the Min-
+imum Vertex Coloring Problem, which is used within our
+proposed hybrid approach. First, let S be a set of all possi-
+ble independent sets in the graph G = (V, E). Also, let bus
+be a binary parameter that holds 1 if the vertex u ∈ V is
+present in the independent set s ∈ S, and 0 otherwise. Fi-
+nally, we associate a binary variable ys to each independent
+set s ∈ S; it takes 1 if the related independent set is selected;
+0 otherwise. Solving the MVCP comes then to solving the
+following extended formulation:
+min
+�
+s∈S
+ys
+(5)
+s.t.,
+�
+s∈S
+busys = 1,
+∀u ∈ V
+(6)
+ys ∈ {0, 1},
+∀s ∈ S
+(7)
+1 Extended formulations are mathematical models in which the num-
+ber of variables grows exponentially as the input increases.
+where (5) is set to minimize the number of selected inde-
+pendent sets, while ensuring that each vertex of the graph
+is present in exactly one of them (see equation (7)). Note
+that, by considering each independent set s ∈ S as a color
+assignment, the adjacency constraints related to the Mini-
+mum Vertex Coloring problem are automatically respected
+(see definition (1)).
+The number of all independent sets in a graph, and hence
+the number of ys variables, can be extremely large, expo-
+nentially growing as the number of vertices in the graph
+increases.
+Hence, as one may anticipate, finding all such
+sets on a given graph is a very hard task and can be very
+time and resource-consuming even for small instances. To
+overcome the aforementioned limitations, we propose a hy-
+brid classical-quantum column generation-based framework
+to efficiently solve the proposed extended formulation by
+enumerating only a small subset of independent sets.
+In
+what follows, we present the related works and discuss how
+a quantum sampler can be integrated into classical frame-
+works.
+II.
+RELATED WORK
+Several quantum algorithms have been proposed for solv-
+ing graph coloring problems in the past years, and most of
+them rely on a quantum annealing-based approach. In [27],
+the authors investigate a real-time quantum dynamics-based
+quantum annealing approach where the related Hamilto-
+nian is designed to naturally respect all problem-related con-
+straints without adding penalty terms. Authors in [28], on
+the other hand, propose a genetic algorithm-based quantum
+approach to solve both vertex and edge coloring problems in
+different highly configurable circuit-based models. Titiloye
+and Crispin [29] compare classical and quantum annealing
+approaches in solving graph coloring problems. According to
+the authors, the path-integral Monte Carlo-based quantum
+annealing (QA) algorithm outperforms its classical counter-
+part.
+Authors in [30] propose another approach in which the
+problem-related set of constraints is transformed into an
+energy minimization problem to output a QUBO formula-
+tion, which is then solved in a quantum annealer platform.
+However, by running various numerical simulations and com-
+paring results obtained with standard and enhanced circuit-
+based QAOA algorithms, authors in [31] indicate the limi-
+tation of the existing QA hardware solutions for solving the
+Vertex Coloring problem. Also, Silva et al [32] compare sim-
+ulated and quantum annealing approaches for solving the
+proposed QUBO formulation for the Vertex Coloring prob-
+lem. Using D-Wave 2X as an independent set sampler for
+a simple greedy framework, the authors show that the pro-
+posed quantum sampler could improve the results with high
+probability on small graphs due to hardware limitations.
+Fabrikant and Hogg [33] introduce a quantum heuristic for
+graph coloring for instances that can be solved with at most
+3 colors. Using two qubits to each vertex of the graph, an
+approximate asymptotic analysis suggests polynomial-time
+cost for solving the related 3-coloring problem. Moreover,
+authors in [34] introduce an exponential-space quantum al-
+gorithm to solve the MVCP in O(1.9140|V|) running time.
+They propose a quantum random access memory framework
+based on Ambainis quantum dynamic programming [35]
+with applications of Grover’s search to branching algorithms.
+
+4
+Moreover, authors in [36] proposed a greedy quantum algo-
+rithm for solving the MVCP by iteratively computing the
+solutions of Maximum Independent Set problems. By sim-
+ulating a framework on a classical computer to reproduce
+the Rydberg blockade phenomenon on neutral atoms-based
+QPUs, they show that their approach can always find fea-
+sible solutions for the problem.
+More details about their
+approach are presented in Appendix A.
+Finally, authors in [37] propose a quantum annealing-
+based method to solve one of the two pricing sub-problems of
+a column generation-based approach for the refinery schedul-
+ing problem.
+They first decompose the problem into one
+master problem and two pricing sub-problems and formu-
+late one of them as a QUBO model. While only the QUBO-
+related pricing sub-problem is solved using a quantum an-
+nealing approach, the master and the other pricing sub-
+problem are solved with a classical solver.
+The quantum
+system is based on D-Wave’s quantum annealers and is de-
+signed to return only the optimal solution.
+Due to some
+serious limitations either related to the problem size or to
+the connectivity of its variables, the authors could guaran-
+tee high-quality solutions for only a few instances of small
+graphs.
+Note that, even though those works propose interesting
+approaches to solving combinatorial problems (sometimes
+only the decision version), little attention has been given
+to hybrid and analog approaches, especially using neutral
+atoms-based QPUs. In what follows, we introduce a com-
+plete quantum pricing framework based on neutral atom
+QPUs that can be easily embedded into a column genera-
+tion algorithm to solve hard combinatorial problems, such
+as graph coloring problems.
+III.
+NEUTRAL ATOM QPUS
+In neutral atom-based QPUs, lasers or microwaves are
+used to induce transitions between electronic states of the
+valence electron of alkali metal (typically Rubidium) atoms.
+Different pairs of electronic levels can be used as qubits.
+Here, we will solely focus on the case where those two states
+are the electronic ground state |g⟩ ≡ |0⟩ and a s− Rydberg
+level |r⟩ ≡ |1⟩. In that case, atoms can be placed arbitrarily
+in space, so that the effective Hamiltonian of the atoms at
+time t can be written as
+H(t) = Ω(t)
+|V|
+�
+u=1
+ˆσx
+u − ∆(t)
+|V|
+�
+u=1
+ˆnu +
+|V|
+�
+u<v=1
+Uuvˆnuˆnv,
+(8)
+where the amplitude (giving the Rabi frequency) Ω(t) and
+detuning ∆(t) of the laser can be controlled over time, and
+the interaction strength Uuv ∝ |ru − rv|−6 is a function of
+the distance between atom u and atom v. Throughout this
+paper, we set ℏ = 1.
+Note that in the current work, we
+consider only a uniform global laser control over the atoms.
+A key property of Rydberg physics is the so-called Ryd-
+berg blockade mechanism [9] : the two-body interaction term
+in (8) forbids the simultaneous excitation of two atoms that
+are closer than a certain distance. Given a set of atoms and
+their positions, one can then define a graph such that each
+atom corresponds to a vertex and in such a way that two
+vertices are connected by an edge if and only if the distance
+between the related atoms is shorter than a given threshold.
+This kind of graphs are known as Unit-Disk graphs, and they
+are the most natural graphs to encode in a neutral atom
+QPU. For this graph class, one can ensure that the evolu-
+tion of the quantum system is restricted to a subspace of the
+complete Hilbert space corresponding to independent sets
+of the graph. By setting Ω(t) = 0 and adjusting the value
+of ∆, one can ensure that the ground-state corresponds to
+an MIS. Because of these properties, people have explored
+Quantum Annealing as a way to solve optimization prob-
+lems on graphs [38]. For non-UD graphs, however, one can
+construct alternative approaches, similar to what is done in
+QAOA, or Variational Quantum Eigensolvers [39].
+IV.
+PROBLEM DECOMPOSITION
+We dedicate this section to fully describing the decompo-
+sition of the proposed extended formulation (5)-(7), a fun-
+damental step to solve the related combinatorial problem
+with a column generation-based algorithm. The need of ap-
+plying such a mathematical strategy comes from the fact
+that, in most of the cases, generating all elements that will
+be related to the variables of an extended formulation is a
+very hard task. For instance, enumerating all independent
+sets of a graph can be impractical even for small instances.
+To overcome the aforementioned issue, we decompose the
+problem under consideration into two problems, named Re-
+stricted Master Problem (RMP) and Princing Sub-Problem
+(PSP). While the former has only a small subset of variables
+needed to find a solution to the problem, the latter is de-
+signed to provide new elements (e.g., independent sets) that
+respect all technical constraints imposed by the RMP. These
+new elements are then added as new variables (also seen as
+columns) to the mathematical model related to the RMP as
+they might potentially improve the quality of the solution
+(e.g., decreasing the number of colors needed to solve the
+Vertex Coloring problem).
+Column generation-based approaches rely on the duality
+theory [40], which states that optimization problems can be
+addressed from two different perspectives: the primal prob-
+lem or its counterpart, the dual problem. The relationship
+between these two problems is the following: (i) for each
+variable (resp. constraint) in the primal problem, there is a
+related constraint (resp. variable) in the dual problem, and
+(ii) the optimization direction (e.g., maximization or mini-
+mization) on the dual is inversed related to its primal coun-
+terpart. For each sub-optimal solution that satisfies all the
+constraints on the primal problem, there is at least one di-
+rection to move in such a way that the objective function is
+improved. Such improving directions are represented by the
+vector of dual variables and optimizing them is equivalent
+to tightening the bounds of the primal problem. For an in-
+depth discussion on the primal-dual relationship, one may
+refer to [40]. After solving a linear model related to a primal
+problem, one can easily get such a direction vector (i.e., dual
+variables) by calling some built-in function proposed by the
+solver used in the process.
+In order to design an efficient column generation frame-
+work, the PSP is then formulated in such a way to incorpo-
+rate the dual information provided by current solutions of
+the RMP, which implies solving a different pricing instance
+each time the sub-problem is called.
+Hence, by applying
+the duality theory, the PSP searches for new variables that
+can improve the objective function of the RMP, and once
+it is mathematically proven that it is no longer possible to
+
+5
+generate such variables (i.e., new columns), the loop-based
+procedure stops. This approach is very powerful when only
+a few variables are normally activated (i.e., taking any value
+other than zero) in the optimal solution to a given combi-
+natorial problem. Hence, applying such a technique, only a
+very small subset of variables is needed to be generated by
+the pricing routine.
+In what follows, we present a decomposition scheme for
+solving the proposed extended formulation for the MVCP.
+The main idea relies on generating a restricted model with
+only a sub-set of independent sets and iteratively updating
+it with new variables (i.e., columns) that have the potential
+to improve the current solution.
+A.
+The restricted master problem
+Since G has an exponential number of potentially suitable
+colorings represented by the related set S of independent
+sets, the extended formulation (5)-(7) admits an exponential
+number of variables. To overcome the difficulty of generating
+S, we propose to generate only a small sub-set S′ ⊆ S of
+variables that are needed to solve the master problem. For
+instance, a trivial solution might be initializing S′ with only
+the singletons of the input graph. The reduced model is then
+hereafter referred to as Restricted Master Problem (RMP),
+and can be defined as follows:
+min
+�
+s∈S′
+ys
+(9)
+s.t.,
+�
+s∈S′
+busys = 1,
+∀u ∈ V
+(10)
+0 ≤ ys ≤ 1,
+∀s ∈ S′
+(11)
+Note that, in order to apply the duality theory, we must
+solve the linear relaxation on the RMP, meaning the y vari-
+ables are no longer binary in this formulation. The RMP is
+then solved again with the integrality constraints (7) once it
+has all variables needed to provide the optimal solution for
+the relaxed RPM, which can be proven mathematically; we
+provide this proof in the following section.
+B.
+Pricing sub-problems
+We present first the dual formulation related to the relaxed
+RMP (9)-(11). By associating a dual variable wu to each
+constraint in (10), we define the dual problem as follows:
+max
+�
+u∈V
+wu
+(12)
+s.t.,
+�
+u∈V
+wubus ≤ 1,
+∀s ∈ S′
+(13)
+wu ∈ R,
+∀u ∈ V
+(14)
+The separation of inequalities (13) represents the pricing
+sub-problems related to the extended formulation (9)-(11).
+The PSP consists then in finding a new coloring set in such a
+way that all adjacency constraints would be respected while
+improving the solution cost of RMP (i.e., decreasing the
+value of the objective function (9)). For this purpose, let
+¯wu be the components of the current dual solution of the
+RMP related to constraints (10). By setting each dual vari-
+able ¯wu as the weight of the related vertex u ∈ V, finding a
+new independent set under such conditions comes to solving
+the following maximum weighted independent set (MWIS)
+formulation:
+max
+�
+u∈V
+¯wuxu
+(15)
+s.t.,
+xu + xv ≤ 1,
+∀(u, v) ∈ E
+(16)
+xu ∈ {0, 1},
+∀u ∈ V
+(17)
+where xu is a binary variable that holds 1 if it is in the in-
+dependent set; 0 otherwise. Inequalities (16) ensures that
+the new independent set respects the adjacency constraints.
+Note that the pricing formulation (15)-(17) is an Integer Pro-
+gramming (IP) version of the QUBO formulation (4).
+The net gain of adding a new variable related to a solution
+to the pricing problem is given by the reduced cost. Based
+on the separation of inequalities (13), we calculate the re-
+duced cost rs for any independent set s given by solving the
+formulation (15)-(17) as following:
+rs = 1 −
+�
+u∈V
+¯wuxu
+(18)
+Since we minimize the master problem in the work, any
+solution with a negative reduced cost might potentially im-
+prove the solution of the RMP. The pricing problem consists
+then in finding a new independent set s whose total weight is
+strictly greater than 1; the total weight of any independent
+set is calculated as in the cost function (15). Hence, any
+solution to the formulation (15)-(17) whose cost is greater
+than 1 can therefore be added to the sub-set S′; if such a
+solution does not exist, then the solution of the RMP can-
+not be improved and, hence, the optimal solution the relaxed
+RPM can be achieved with the current variables related to
+the sub-set S′.
+V.
+SOLVING METHOD
+As previously discussed, solving the pricing sub-problems
+is usually the bottleneck in column generation-based algo-
+rithms since it comes to solving different instances of a hard
+combinatorial problem multiple times.
+To overcome this
+problem, we propose a quantum pricing algorithm that can
+find the (near-) optimal solution faster than the classical
+one (i.e., where no QPU is involved). For this purpose, let
+us now describe the column generation-based framework pro-
+posed to solve the Minimum Vertex Coloring problem; which
+is summarized in Fig. 3.
+First, a minimal sub-set S′ ⊆ S of independent sets is
+generated in such a way that it ensures a feasible solution for
+the extended formulation (5)-(7). As previously discussed,
+the most trivial way to build the initial set S′ of independent
+
+6
+Create the initial set S
+of possible independent
+sets (IS) that give a
+feasible solution
+Generate the
+Reduced Master
+Problem (RMP)
+Solve the relaxed
+RMP with the current
+set S 
+Get the dual
+variables 
+Generate the Pricing
+Sub-Problem (PSP)
+with the dual
+variables
+Solve the PSP
+Update S with the
+new set(s) found by
+PSP  
+Solve the 0-1 RMP
+with the final set S 
+No
+Yes
+Stop 
+Criteria ?
+Problems's 
+instance
+Final coloring  
+solution
+FIG. 3: Interaction between the restricted master problem
+and the sub-problem
+sets is generating only the singletons in the graph; this simple
+approach always provides a solution for the RMP.
+The classical part of the proposed hybrid approach is re-
+lated to the Restricted Master.
+Once the initial set S′ is
+created, the RMP is built and then solved on its linear re-
+laxation form (see formulation (9)-(11)) by a classical solver
+(e.g., GPLK). The values of the dual variables are also given
+by the classical solver by running a built-routine after solv-
+ing each version of the RMP (i.e., with different sub-sets of
+variables).
+The next steps are related to the pricing sub-problems,
+in which the PSP is solved by applying the values of the
+related dual variables from the solved RMP. As previously
+discussed, this step comes to finding independent sets whose
+weight is strictly greater than 1. If such elements exist, then
+they are added to S′.
+As we detail in the following, we
+propose a quantum sampler that is specifically tailored to
+output multiple independent under the aforementioned con-
+ditions For each new independent set found by solving the
+related pricing sub-problem, a new variable is created and
+added to the sub-set S′.
+Then, the RMP is solved again
+with the new columns (i.e., independent sets converted into
+variables).
+These last steps are repeated until no column
+is generated by the PSP. Finally, the final RMP is solved
+with all generated variables (i.e. independent sets) with the
+integrality constraints (7), as previously discussed.
+A.
+Worked example
+Table I shows a worked example of applying the column
+generation framework on the graph represented in the Fig. 4,
+where the RMP is solved classically by an IP solver. The
+first column shows how many PSPs were solved before reach-
+ing the final solution. The second column depicts the inde-
+2
+3
+1
+4
+5
+FIG. 4: Illustration of a graph with 5 vertices and 3 edges;
+the set S of all independent sets is composed by the five
+singletons and the sub-sets [1,2], [1,4], [2,3], [2,4], [2,5], [3,5],
+[4,5], [1,2,4], [2,3,5], [2,4,5].
+pendent sets selected as the solution for the RMP formula-
+tion (9)-(11). The third column presents the dual solution
+given by the variables w (see the dual formulation (12)-(14)).
+Finally, the last column depicts the maximum weighted inde-
+pendent sets generated after running each PSP, where the ¯w
+parameters in the cost function of the formulation (15)-(17)
+are set to the values of dual variables w; the generated
+weighted independent set is then added to the set S′ be-
+fore rerunning the RMP. The final coloring is represented by
+the last solution for the RMP, where one color is given for
+each selected independent set.
+In this example, we generate all five singletons of the graph
+as the first sub-set S′ of independent sets before solving the
+first version of the RMP. As observed, only 4 more inde-
+pendent sets (out of the remaining 10 to be generated) were
+needed to find the best solution. Indeed, even though the
+closed-loop could be stopped after iteration 4, the optimality
+was only proven after the fifth iteration, when no indepen-
+dent set whose total weight is greater than 1 can be gener-
+ated (see the dual solution on the last row, which is used as
+vertex weights). Note that the independent set generated in
+the i-th iteration is likely to be selected in the solutions of
+the RPM in the following iteration.
+The values of the dual variables can be calculated by solv-
+ing the dual formulation (12)-(14), where the cost function
+is set to be equal to the solution cost of the RMP in the
+same interaction. In this example, the cost function of the
+dual formulation was set to be equal to 5, 3, 3, 2, and 2 in
+the first, second, third, fourth, and fifth iterations. How-
+ever, most commercial solvers solve the dual formulation in
+parallel in order to prove the optimality of the (RMP) pri-
+mal solution.
+Hence, the final value of the dual variables
+can then be easily provided by a built-in solver’s function
+(e.g., solver.get dual()). Note, however, that only one inde-
+pendent set (the most weighted one) is generated by solving
+the related pricing formulation (15)-(17) and, due determin-
+istic nature of this approach, the final solution is always the
+same if the input (i.e., the graph and vertex weights) does
+not change.
+Now, let us exemplify how a quantum sampler could be
+applied to solve the pricing sub-problems within the column
+generation framework; Table II shows such a worked exam-
+ple. As presented in the previous worked example depicted
+in Table I, we consider the graph depicted in Fig. 4 initial
+sub-set S′ composed only of singletons. As observed, only
+three iterations are needed to find the final solution. Indeed,
+sampling more independent sets on each pricing iteration can
+considerably improve the performance of the column gener-
+ation algorithm. In this example, 9 out of 10 possible inde-
+pendent sets were generated. In fact, due to its stochastic
+nature, a quantum sampler can efficiently provide multiple
+sets with the same input and, hence, speed up the conver-
+Iteration
+RMP solution
+Dual solution
+MWIS
+1
+[1] , [2] , [3] , [4] , [5] [1.0, 1.0, 1.0, 1.0, 1.0] [1, 2, 4]
+2
+[3] , [5] , [1, 2, 4]
+[1.0, 1.0, 1.0, -1.0, 1.0] [2, 3, 5]
+3
+[3] , [5] , [1, 2, 4]
+[1.0, -1.0, 1.0, 1.0, 1.0]
+[1, 4]
+4
+[1, 4] , [2, 3, 5]
+[1.0, -1.0, 1.0, 0.0, 1.0]
+[3, 5]
+5
+[3, 5], [1, 2, 4]
+[1.0, 0.0, 1.0, 0.0, 0.0]
+None
+TABLE I: A worked example of applying the column gen-
+eration framework on the graph represented in the Fig. 4.
+Only 5 independent sets (out of 10) are generated to find
+the optimal solution.
+
+7
+Iteration RMP solution
+Dual solution
+IS generated
+1
+[1],[2],[3],[4],[5]
+[1.0, 1.0, 1.0, 1.0, 1.0] [1, 2, 4],[2, 4, 5]
+[2, 3, 5] , [1, 2]
+[2, 5] , [2, 3]
+[4, 5] , [3, 5]
+2
+[3, 5], [1, 2, 4]
+[1.0, -0.5, 1.0, 0.5, 0.0]
+[1, 4]
+3
+[1, 4],[2, 3, 5]
+[1.0, 0.0, 1.0, 0.0, 0.0]
+None
+TABLE II: A worked example of applying a quantum sam-
+pler to solve the pricing sub-problems related to the column
+generation framework. Considering the graph represented in
+the Fig. 4, only 3 different pricing sub-problems were solved
+before reaching the final solution.
+gence of the RMP to the optimal solution.
+In what follows, we present how neutral atom-based sys-
+tems can be designed to take into consideration the dual val-
+ues from the RMP to output multiple weighted independent
+sets and efficiently solve the related PSPs.
+B.
+Embedding strategy
+Embedding random graphs into UD-disk representations
+is proven to be an NP-hard problem [41]. Also, not all graphs
+have such a realization. For instance, it is not possible to find
+a UD-disk representation for any K1n star graph with n > 6
+on a 2-dimension plane. In order to design a near-optimal
+register to represent any graph given as an input, we use
+the embedding strategy presented in [3], where an algorithm
+based on force-directed principles is used to embed graphs
+into planes in such a way that two connected (resp. dis-
+joint) vertices are placed close to (resp. far from) each other,
+with a minimum (resp. maximum) distance between them
+(resp. from the plane’s center). See Appendix B for more
+details.
+Once the initial register is created as previously discussed,
+it has the same number of atoms as the number of vertices
+on the initial graph G.
+However, solving the pricing sub-
+problem within the column generation framework might po-
+tentially imply finding one or more independent sets on a
+sub-graph G′: this sub-graph is generated with only the ver-
+tices with positive weight (i.e., the positive dual value pro-
+vided by the solved restricted master problem). The light
+pattern holding the atoms in place is created via a spatial
+light modulator, and determining the right settings for this
+device demands lengthy calibrations [20].
+Therefore,
+creating
+a
+new
+register
+by
+running
+the
+Algorithm1 Vertex-atom remapping
+Input: A graph G′, a register R and vertex weights W.
+Output: The reduced register R.
+1: Let ¯V′ be the list of vertices from G′ sorted in descending order by
+the weight given by W
+2: Remove all vertices whose weights are less than or equal to zero
+from ¯V′
+3: Map vertex ¯V′.first to the furthest position from the center of R
+4: ¯V′ ← ¯V′\¯V′.first
+5: while ¯V′ has vertices’ do
+6:
+Map vertex ¯V′.first to the furthest position from all vertices
+already mapped to an atom in R
+7:
+¯V′ ← ¯V′\¯V′.first
+8: end while
+9: Remove all atoms not mapped to any vertex from register R
+10: return register R
+(a) Sub-graph G′.
+(b) Initial register.
+(c) Vertex-atom mapping.
+(d) Final register.
+FIG. 5: Vertex-atom mapping for a 5-vertex graph on a 9-
+atom register. Fig. 5a presents a graph with 5 vertices, whose
+weights are to their indexes (e.g., the weight of vertex 1
+(resp. 5) is equal to 1 (resp. 5)). Fig. 5b shows a register
+with 9 atoms in their positions. Fig. 5c depicts a mapping
+where the most weighted vertices are far from each other.
+Finally, Fig. 5d presents the final register and the vertices
+from the sub-graph they represent.
+Fruchterman-Reingold algorithm with the remaining vertices
+would be too time-consuming. Instead of just removing the
+atoms mapped to the vertices that are no longer in the sub-
+graph G′, we propose here a vertex-atom remapping strategy
+that takes into consideration the vertices’ weights (dual val-
+ues).
+Algorithm 1 summarizes the main idea of our proposed
+approach: it receives a sub-graph G′ = (V′, E′), a register R
+composed of atoms and their positions within the QPU, and
+vector W representing the weights for all vertices in V′. First,
+let ¯V′ be the list of vertices of G′ sorted in descending or-
+der by the weight given by W. Then, remove all vertices
+whose weights are less than or equal to zero from ¯V′ (step 2),
+map the most weighted vertex to the further atom from the
+register’s center (step 3), and remove the remapped vertex
+from ¯V′ (step 4). Then, the most weighted vertex in ¯V′ is
+embedded into the furthest atom position from all atoms al-
+ready mapped to a vertex in G′, and removed from ¯V′; these
+steps are done until all vertices from G′ are mapped to an
+atom in the register R. Finally, as seen in step 9, all atoms
+not mapped to any vertex are removed from R. Let us re-
+call that, since the proposed remapping strategy is applied
+to solve the PSPs, the weight vector W is generated with
+dual values provided by the solution of the current RMP,
+as previously mentioned. A worked example of applying the
+proposed algorithm is depicted in Fig. 5.
+C.
+Independent set quantum sampler
+We propose an independent set sampler based on neutral
+atoms, which is inspired by the quantum adiabatic approach
+described by [42]. In the latter, the main idea is to slowly
+evolve the system from an easy-to-prepare ground state to
+the ground state of the final cost Hamiltonian HC. By slowly
+evolving the system, the atoms stay in the instantaneous
+ground state [43]. Here, we only aim at keeping the system
+close to the instantaneous ground state, allowing it to pick
+up components in the low-lying states.
+In order to do such evolution, we continuously vary the
+detuning δ(t) and the Rabi frequency Ω(t) in time, starting
+with Ω(t0) = 0, δ(t0) < 0 and ending with Ω(tf) = 0 ,
+
+3
+5
+2
+42
+4
+3
+54
+2
+3
+58
+δ(tf) > 0. Hence, the initial ground-state of H(t0) is |00000
+�
+,
+while the low-energy space of H(tf) contains that of the cost
+Hamiltonian HC. This protocol originally aims at preparing
+the ground-state and solving the MIS problem. In this work,
+we use it to sample independent sets whose total weight is
+equal or close to that of the MWIS.
+To ensure that we are not exciting the system to states
+that do not form independent sets, we have to estimate the
+minimal distance between atoms that are disjoint in the
+graph (this yields Ωd), and estimate the furthest distance
+between two connected atoms, which gives Ωc. For this pur-
+pose, let duv be the distance between nodes u and v from V,
+while C6 is the interaction coefficient related to the quan-
+tum device.
+Then, Ωd and Ωc are respectively defined as
+following:
+Ωd = argmax
+(u,v)/∈E
+C6d−6
+uv
+(19)
+Ωc = argmin
+(u,v)∈E
+C6d−6
+uv
+(20)
+In unit-disk (UD) graphs, keeping Ω ∈ [Ωd, Ωc] ensures
+that only independent sets appear in the dynamics (provided
+that the dynamics is not too fast) [38, 44]. A large value of
+Ω is desirable to speed up the dynamics of the system and
+reach states that are far from the initial state. That is why
+we set Ωmax = max(Ωc, Ωd) as the maximum value the Rabi
+frequency can take during the pulse sequence.
+In the case of non-UD graphs, this value of Ωmax is still
+a good compromise. Indeed, if Ω < Ωc then the effective
+graph G′ that is encoded in the system contains more edges
+than the target graph G. The independent sets of G′ are then
+not strictly included in the independent sets of G, and the
+pricing may then miss some variables. On the other hand,
+if Ω > Ωd, all edges of G′ are also edges of G, and therefore,
+the independent sets of G′ are strictly included in the inde-
+pendent sets of G. This ensures that the pricing is still able
+to explore all independent sets and therefore provide new
+variables to improve the current solution of the RMP.
+As inputs, the quantum sampler gets an atom register and
+the pulse representing the designed Hamiltonian. Then, af-
+ter running the proposed neutral atom-based quantum algo-
+rithm, different outputs are possible:
+1. Return only the largest independent set;
+2. Return all independent sets;
+3. Return all independent sets whose weights are greater
+than a given threshold;
+4. Return only the most weighted independent set;
+For applying the two last strategies, one must also provide
+a weight vector and a cost function as inputs to the quantum
+sampler. In this work, we apply the third output case men-
+tioned above by applying the cost function (15) to qualify
+any independent set; note that for unweighted graphs, this
+cost function can be applied by setting all weights to 1.
+D.
+Solving the Pricing sub-problems
+Let ¯w be the vector of dual values related to the solved
+relaxed RMP as previously discussed.
+Then, we generate
+the graph G′ = (V′, E′, ¯w) in such a way that V′ = {u ∈
+V| ¯wu > 0} and E′ = {(u, v) ∈ E| ¯wu ¯wv > 0}. Also, for each
+vertex u ∈ V′, the related weight is given by ¯wu ∈ ¯w.
+We run the quantum algorithm previously described on
+the related graph G′ = (V′, E′, ¯w), where ¯w is the dual vec-
+tor from the current solve solution for the primal problem.
+While we keep only the atoms related to the vertices in V′,
+their positions might potentially be permuted as described
+in Algorithm 1. Also, the pulse shape might be adjusted to
+the new register by calculating the new Ωmax, which is done
+by calculating the distance between each pair of qubits, as
+previously proposed. By applying the objective function (15)
+to qualify each bitstring sampled by the QPU, the S′ is then
+updated with all sampled weighted independent sets whose
+total weight is strictly greater than 1. Let us recall that the ¯w
+might potentially be a vector composed of very small values
+(i.e., ¯wu ≪ 1, ∀u ∈ V′, including zero and negative values),
+meaning that any independent set, including the maximum
+one, might have a total weight strictly lower than 1: in this
+case, no independent set is added to S′.
+E.
+Stopping criterion
+The algorithm stops the inner loop when no column is
+generated after solving the related pricing sub-problem. This
+means that either the current sub-set S′ is already composed
+of all independent sets needed to find the optimal for the
+relaxed formulation (9)-(11) (if the PSP is exactly solved), or
+the PSP solver does not find any solution under the imposed
+constraints (if the PSP is solved heuristically).
+Once the
+inner loop stops, the ILP version of the final RMP is solved
+with all generated columns (i.e., variables/independent sets)
+and the integrality constraints (7).
+It is important to notice that the final solution given by
+the related LP formulation (9)-(11) might not be the same
+(or even the optimal one) for the ILP formulation (5)-(7)
+(i.e., with the integrality constraints (7)).
+To ensure op-
+timality, the proposed Hybrid Column Generation should
+be embedded into a Branch-and-Price framework. This ap-
+proach, however, is out of the scope of this work. For more
+information, one may refer to [1].
+VI.
+NUMERICAL SIMULATIONS
+Let us first describe the setup used in our numerical sim-
+ulations.
+While random graph instances were generated
+with the Vladimir-Brandes algorithm [45], which produces
+Erd˝os–R´enyi graphs, UD-guaranteed graphs were produced
+as proposed in [46]. It is worthwhile to mention that random
+graphs are unlikely to be unit-disks. Indeed, the probabil-
+ity of having a UD Erd˝os–R´enyi graph quickly approaches
+zero as the number of the vertices increases and the den-
+sity remains stable. For this reason, this graph class (i.e.,
+random graphs) is hereafter referred to as non-UD graphs.
+For each graph class, we generated 30 instances for three
+different graph densities by setting the probability p of con-
+necting any pair of vertices with an edge to 20%, 50%, and
+80%. Also, the optimal solution of each instance was found
+by exactly solving the compact formulation of the Minimum
+Vertex Coloring problem [47].
+We compare our proposed hybrid algorithm to several
+state-of-the-art approaches, both classical and quantum. In-
+deed, it is possible to solve the pricing problem (15)-(17)
+
+9
+4
+6
+8
+10
+12
+14
+number of vertices
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+3.2
+3.4
+number of iterations
+Strategy
+AR
+AIPR
+AR-HRD
+AIPR-HRD
+(a) 0.2-density,UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2.0
+2.2
+2.4
+2.6
+2.8
+number of iterations
+(b) 0.5-density,UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2.0
+2.1
+2.2
+2.3
+2.4
+2.5
+2.6
+number of iterations
+(c) 0.8-density,UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+number of iterations
+(d) 0.2-density, non-UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2
+3
+4
+5
+6
+number of iterations
+(e) 0.5-density, non-UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+3.2
+3.4
+3.6
+number of iterations
+(f) 0.8-density, non-UD graphs.
+FIG. 6: Number of iterations on the QPU emulator before reaching the stop criteria on different graph classes (UD and
+non-UD), orders (from 4 up to 14 vertices), and densities (20%, 50%, and 80% of all possible connections) and applying
+different register and Hamiltonian redesign strategies: only atom removal (AR), atom index permutation and atom removal
+(AIPR), atom removal with Hamiltonian redesign (AR-HDR), and AIPR with Hamiltonian redesign (AIPR-HDR).
+with a classical solver, where G′ is given as a vertex-weighted
+graph. Note, however, that only the optimal weighted inde-
+pendent set is generated on each iteration. As previously
+discussed, due deterministic nature of this approach, here-
+after referred to as Classical CG, the final solution does not
+change same if the input (i.e., the graph and vertex weights)
+remains the same. We also compared our proposed approach
+to the Classical Greedy algorithm described in Appendix A,
+where a maximum independent set was provided by a classi-
+cal solver on each iteration. We use GLPK [48] as the linear
+solver to solve the (reduced) master problem and classical
+pricing sub-problems.
+In order to compare our neutral atom-based quantum
+sampler to other stochastic approaches, we implemented a
+greedy generator, hereafter referred to as Greedy CG, that
+can randomly generate multiple weighted independent sets.
+For more details, see Appendix C. We also compared our
+approach to a simulated annealing (SA)-based solver, here-
+after referred to as SA CG, where we minimize the related
+maximum weighted independent set QUBO matrix described
+in (4) with the classical D-Wave QUBO sampler [49]; the
+weights are given by the dual values of the solved RMP on
+each iteration, while α is set to the sum of absolute values
+of weights.
+For all stochastic sub-routines, the maximum
+number of tries was set to 1000. Moreover, several pricing
+iterations can be done before getting an improvement on the
+RPM solution. This is due to the inherent symmetry of the
+solution space related to the instance of the problem, driv-
+ing the algorithm to generate independent sets with the same
+cost. In all CG-based approaches, the maximum number of
+pricing iterations allowed without any improvements in the
+RMP solution was set to 3. Finally, we also compared our
+hybrid approach against a quantum version of the greedy
+algorithm described in Appendix A, where only the largest
+independent set sampled by the proposed quantum sampler
+described in section V C is returned on each iteration. This
+approach is hereafter referred to as Quantum Greedy and is
+based on the work proposed by Vitali et al [36].
+The register related to each pricing sub-problem was cre-
+ated by applying the embedding strategy presented in Sec-
+tion V B. To this end, each atom’s position was found with
+the spring layout function from Networkx package [50]: we
+multiplied each position vector by 40 in order to respect the
+distance constraints imposed by the device. Moreover, we
+applied the Hamiltonian design strategy described in section
+V C.
+The evolution of the quantum system under the predicted
+pulse for a given register was then simulated using Pulser’s
+simulation module [51].
+Both noiseless and noisy simula-
+tions were performed. Noiseless simulations involve solving
+the time-dependent Schr¨odinger equation. The output of a
+noiseless simulation is a vector in the Hilbert space that can
+be sampled a finite number of times in order to mimic a real
+experimental setup with a limited measurement budget. In
+order to assess the robustness of our approach against noise,
+noisy simulations were also performed. These calculations
+are numerically expensive and we restrict our study to State
+Preparation And Measurement (SPAM) errors. These are
+expected to be the main source of noise and can be obtained
+by post-processing of noiseless results (for more details, see
+[39]). We set here the noise parameters to realistic values
+based on current hardware specifications.
+They are sum-
+SPAM
+bad preparation η
+0.005
+false positive ϵ
+0.03
+false negative ϵ′
+0.08
+Temperature
+30µK
+Laser waist
+148µm
+TABLE III: Noise parameters used in noisy emulations.
+
+10
+4
+6
+8
+10
+12
+14
+number of vertices
+2
+4
+6
+8
+10
+number of iterations
+Approach
+Classical CG
+Classical Greedy
+Quantum Greedy
+Greedy CG
+SA CG
+Noiseless Quantum CG
+(a) 0.2-density, UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2
+4
+6
+8
+10
+12
+number of iterations
+(b) 0.5-density, UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2
+4
+6
+8
+10
+number of iterations
+(c) 0.8-density, UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2
+4
+6
+8
+10
+number of iterations
+(d) 0.2-density, non-UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2
+4
+6
+8
+10
+12
+number of iterations
+(e) 0.5-density, non-UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+2
+4
+6
+8
+10
+12
+number of iterations
+(f) 0.8-density, non-UD graphs.
+FIG. 7: Number of iterations before reaching the stop criteria on different graph classes (UD and non-UD), orders (from 4
+up to 14 vertices), and densities (20%, 50%, and 80% of all possible connections) by applying different approaches: Classical
+Column Generation (CG), Greedy CG, SA CG, Quantum CG, Classical Greedy, and Quantum Greedy.
+marized in Table III. The register and Hamiltonian design
+were done as previously presented in sections V B and V C,
+respectively.
+A.
+Register and pulse redesign strategies
+We first analyze the impact of four different registers and
+pulse (i.e., Hamiltonian) redesign strategies on the perfor-
+mance of our proposed hybrid classical-quantum column gen-
+eration approach: while the Atom Removal (AR) strategy
+only removes the atoms whose related vertex’s weight is
+equal to or less than zero, Atom Index Permutation and
+Atom Removal (AIPR) strategy also apply the proposed
+vertex-atom remapping algorithm (see Algorithm 1). In ad-
+dition to these two strategies, we tested the Atom Removal
+with Hamiltonian Redesign (AR-HDR), in which the new
+maximal Rabi frequency is recalculated after applying the
+AR strategy.
+Finally, in Atom Index Permutation, Atom
+Removal and Hamiltonian Redesign (AIPR-HDR) strategy,
+the new maximal Rabi frequency can also be recalculated
+after applying the AIPR strategy.
+Fig. 6 shows the average number of pricing iterations (and
+the standard deviation with 95% confidence interval) before
+reaching the stop criteria on different graph classes (UD and
+non-UD), order (from 4 up to 14 vertices), and densities
+(20%, 50%, and 80% of all possible connections).
+As ob-
+served, only a few calls to the quantum sampler (i.e., pricing
+iterations) were needed to reach the best solution of the re-
+laxed version2 of the master problem related to each graph
+2 Let us recall that the dual variables can be only generated from linear
+programs, meaning that the integrality constraints (7) are replaced
+by constraints (11).
+instance. Indeed, the average number of iterations on non-
+UD (resp. UD) graphs was always less than 6 (resp. 3); as
+seen in Fig. 6c (resp. Fig. 6f), only 2 (resp. 3) sampling pro-
+cesses were done in average to solve UD (resp. non-UD)
+graphs with 4 (resp. 13) vertices and 80% density when
+AIRP (resp. AR-HDR) strategy was applied. Even though
+non-UD graphs seem to be more difficult to be solved, es-
+pecially those with 50% of density (see Fig. 6e), applying
+different register and pulse redesign strategies could speed
+up the solving process. While we do not observe any signif-
+icant impact from redesigning the pulse after only removing
+useless atoms from the register (see blue and green lines re-
+spectively related to AR and AR-HDR strategies), recalcu-
+lating the maximum value of the Rabi frequency after per-
+muting atoms’ indices (i.e., applying AIPR-HDR approach)
+could decrease the number of sampling processes by 44% (see
+Fig. 6d). Indeed, as seen in Figures 6a and 6d, this approach
+had the best overall performance, having a stronger impact
+on sparse graphs.
+B.
+Quantum and classical approaches
+We now compare the number of iterations needed to be
+run on different graph classes, orders, and densities by ap-
+plying different approaches: Classical Column Generation
+(CG), Greedy CG, SA CG, Noiseless Quantum CG, Classical
+Greedy, and Quantum Greedy. We applied the AIPR-HDR
+strategy for redesigning each pricing sub-problem within the
+Quantum CG framework.
+While this indicator refers to
+how many times the PSP was solved within both classical
+and quantum column generation frameworks for coloring a
+given graph, it indicates how many independent sets were
+generated during the while-loop on Algorithm 2 by using
+both classical and quantum methods as previously discussed.
+Fig. 7 shows the average number of iterations and the stan-
+
+11
+4
+6
+8
+10
+12
+14
+number of vertices
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+gap
+Approach
+Classical CG
+Classical Greedy
+Quantum Greedy
+Greedy CG
+SA CG
+Noiseless Quantum CG
+(a) 0.2-density, UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+gap
+(b) 0.5-density, UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+gap
+(c) 0.8-density, UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+gap
+(d) 0.2-density, non-UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+gap
+(e) 0.5-density, non-UD graphs.
+4
+6
+8
+10
+12
+14
+number of vertices
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+gap
+(f) 0.8-density, non-UD graphs.
+FIG. 8: Gab between the best solution found and the optimal one on different graph classes (UD and non-UD), orders
+(from 4 up to 14 vertices), and densities (20%, 50%, and 80% of all possible connections) by applying different approaches:
+Classical Column Generation (CG), Greedy CG, SA CG, Noiseless Quantum CG, Classical Greedy, and Quantum Greedy.
+dard deviation (with 95% confidence interval) on 30 graphs
+randomly generated as presented above.
+First, we observe that the Quantum Greedy approach has
+the same overall performance as its classical counterpart,
+showing that our quantum sampler can solve the Maximum
+Independent Set problem efficiently. Also, both strategies
+have the same linear behavior related to the size of the graph,
+i.e., the number of edges it contains, being most impacted
+by dense graphs (see Figures 7c and 7f). This behavior is
+expected since the size of each independent set gets smaller
+as the set of edges gets larger. Hence, more iterations have,
+in general, to be done to cover all vertices of a dense graph.
+Also, while outperforming the Classical CG approach on al-
+most every graph class (in terms of the number of iterations),
+both Classical and Quantum Greedy algorithms had their
+performance slightly decreased on UD graphs. Finally, tak-
+ing advantage of the related superposition aspect, the pro-
+posed Quantum CG outperformed all other approaches on
+all graph classes. For instance, while the Quantum CG algo-
+rithm needed less than 4 (resp. 6) sampling iterations for all
+sparse and dense (resp. 0.5-density) non-UD graphs, Quan-
+tum and Classical Greedy approaches (resp. Classical CG
+algorithm) needed up to 10 (resp. 12) pricing interactions to
+solve the same graph class.
+Fig. 8 shows the average gap (and the standard devia-
+tion with 95% of confidence interval) between the optimal
+solution and the best one found by applying the presented
+approaches. First, we observe that our proposed Quantum
+CG approach has the best overall performance. Indeed, it
+could find the optimal solution in almost all instances; as
+seen in Fig. 8f, our approach could not find the best solu-
+tion only for some 13-vertex non-UD graphs. Also, unlike
+all other approaches, the Quantum CG is not impacted by
+the graph class; while the Classical CG could better perform
+on dense graphs (see Figures 8c and 8f), both Classical and
+Quantum Greedy approaches are more stable on UD graphs.
+Also, as depicted in Fig. 8d (resp. Fig. 8a), the proposed
+Quantum CG algorithm could reduce the average gap on 12-
+vertex non-UD (resp. 13-vertex UD) graphs from roughly
+19% (resp.
+11%) to 0% when compared to the Quantum
+Greedy (resp. Classical CG) approach.
+Our proposed quantum pricing-based approach also out-
+performed both stochastic classical approaches in most of the
+instances, especially those related to UD and sparse graphs.
+Even though the quality of the solutions remains the same
+(see Fig. 8), the Noiseless Quantum CG could reduce by 50%
+the number of iterations on sparse graphs when compared to
+SA-based pricing, as observed in Figures 7a and 7d). Even
+though the Greedy CG has fewer pricing iterations on some
+graph classes, as in bigger non-UD graphs with 20% and 50%
+of density (see Figures 7d and 7e), the average gap could be
+reduced by 80% when our proposed Quantum CG was ap-
+plied on the same graph classes, as we observe in Figures 8a
+and 8d. This indicates that random sampling to find inde-
+pendent sets cannot solve pricing sub-problems effectively.
+C.
+Noisy and noiseless simulations
+We now present the results from our numerical simula-
+tions wherein emulated noise was added as previously dis-
+cussed. Fig. 9 depicts the number of iterations before reach-
+ing the stop criteria and the gap between the final solu-
+tion and the optimal one on different graph orders (from
+4 up to 14 vertices) by applying different approaches: Clas-
+sical CG, Greedy CG, SA CG, Noiseless Quantum CG, and
+Noisy Quantum CG. The results shown are related to ap-
+plying a given approach on all graph instances of the same
+order, regardless of their density and whether they are unit
+disks. First, as seen in Fig. 9a, no impact was observed when
+noise is added to the quantum independent set sampler. In-
+deed, the overall final gap was similar to the noiseless model,
+
+12
+4
+6
+8
+10
+12
+14
+number of vertices
+0.0
+0.1
+0.2
+0.3
+0.4
+gap
+(a) Gab between the final solution and the optimal one.
+4
+6
+8
+10
+12
+14
+number of vertices
+2
+4
+6
+8
+10
+number of iterations
+Approach
+Classical CG
+Noisy Quantum CG
+Greedy CG
+SA CG
+Noiseless Quantum CG
+(b) Number of iterations before reaching the stop criteria.
+FIG. 9: Number of iterations before reaching the stop cri-
+teria and the gap between the final solution and the op-
+timal one on different graph orders (from 4 up to 14 ver-
+tices) applying different approaches: Classical Column Gen-
+eration (CG), Greedy CG, SA CG, Noiseless Quantum CG,
+and Noisy Quantum CG
+also outperforming the classical CG. Finally, as observed in
+Fig. 9b the number of iterations needed to find the optimal
+solution of the relaxed RMP was increased by only 6% in
+average when the noisy model is compared to the noiseless
+one, even on big graphs.
+CONCLUSION
+In this study, we demonstrated that it is possible to incor-
+porate quantum elements into classical state-of-the-art algo-
+rithms in order to improve their performances. Compared
+to the classical column generation, our neutral atom-based
+quantum pricing could reduce by up to 83% the number
+of iterations needed to solve the Minimum Vertex Color-
+ing problem. Also, our proposed hybrid approach could re-
+duce the average gap from 19% to 0% when compared to
+both classical and greedy approaches. Moreover, unlike the
+deterministic approaches, our quantum pricing-based col-
+umn generation is robust to all graph classes, including non
+unit-disk graphs of all tested orders and sizes.
+This indi-
+cates that the proposed hybrid algorithm can efficiently solve
+other combinatorial problems (e.g., Minimum Edge Color-
+ing and Minimum Clustering problems) after some trivial
+translation-based pre-processing. The proposed framework
+also outperformed other stochastic algorithms, especially on
+sparse graphs. Indeed, our quantum pricing-based column
+generation could reduce by up to 50% the number of pricing
+iterations and by up to 80% the gap of the final solution
+on graphs when compared to the simulated annealing-based
+and greedy-based pricing approaches, respectively. Finally,
+we observed that our proposed hybrid framework is robust
+to noise. Indeed, the quality of the final solutions was not
+impacted when compared to the noiseless model, which is
+a great indication of how noise-resilient the analog mode of
+operation can be. Our proposed approach can readily be im-
+plemented on neutral-atom quantum computing hardware.
+Let us recall that our proposed algorithm remains a
+heuristic approach to solving combinatorial problems. Even
+though it can provide high-quality solutions to a plurality
+of instance classes, embedding this method into a Branch-
+and-Pricing framework is necessary to guarantee optimality
+to any instance of the problem under consideration. Since
+providing the initial sub-set of variables for the reduced mas-
+ter problem is an important step in any column generation-
+based algorithm, the proposed quantum sampler can be used
+as a warm starter to generate such a subset of elements (e.g.,
+independent sets). In the case of solving the Minimal Ver-
+tex Coloring problem, by setting all vertex weights to 1, this
+approach might be useful in scenarios where QPU resources
+are limited. Also, this strategy might potentially speed up
+both classical and quantum column generation approaches.
+Moreover, an optimal control-based strategy to design pulses
+to each input instance might potentially be applied to solve
+non-trivial pricing sub-problems. Similarly, different register
+embedding approaches can be developed to take into consid-
+eration different information from the input data.
+ACKNOWLEDGMENTS
+We thank Julien Bernos and Vincent Elfving for insightful
+discussions. During the completion of this manuscript, we
+became aware of a related work [52].
+Appendix A: A Greedy algorithm for the Minimum
+Vertex Coloring problem
+We present here a greedy heuristic based on the Minimum
+Vertex Coloring framework introduced by [36]. The main
+idea relies on interactively solving the Maximum Indepen-
+dent Set problem with only a subset of the vertices of a given
+graph. Algorithm 2 summarizes the proposed approach.
+As input, the Algorithm 2 receives a graph G = (V, E)
+Algorithm2 Greedy Algorithm
+Input: A Graph G = (V, E) and a set C of available colors.
+Output: Color-vertex assignment.
+1: G′ ← G
+2: c ← 1
+3: while G′ has vertices do
+4:
+Find a (maximum) independent set IS in G′
+5:
+Assign color c ∈ C to all vertices from IS in G
+6:
+Remove all incident edges of each vertex in IS from G′.
+7:
+Remove the set IS of vertices from G′.
+8:
+c = c + 1
+9: end while
+
+13
+and a set C of available colors. Note that, to always have
+a feasible color assignment, |V| ≤ |C| must hold. First, a
+copy of G is made with an auxiliary graph G′ ≡
+G.
+A
+variable k is also created: it keeps the index of the first
+available color.
+Then, steps 3-7 are done until no vertex
+remains in G′. On each iteration, a feasible independent set
+in G′, potentially a maximal one, is generated. For instance,
+a classical solver can be used to solve the formulation (4)
+or one may use the proposed quantum sampler described
+in section V C by setting the algorithm to output only the
+largest sampled independent set. Then, all vertices of G with
+the same indices as those in the found independent set IS
+are colored with the first available color from C, whose index
+is given by the variable c. Next, all vertices of IS (and the
+related incident edges) are removed from G′, and then the
+reference of the first available color is updated (see step 8).
+It is worth mentioning that, while the number of qubits
+needed to run this algorithm on a QPU is reduced when com-
+pared to other approaches, its performance can be strongly
+impacted by the order in which the independent sets are gen-
+erated. Also, in the worst case, |V| iterations can be done
+in the quantum device (take a complete graph as an exam-
+ple) before the final solution is found. More details about its
+performance are presented in Section VI.
+Appendix B: Force-directed algorithm
+As pointed in [3], algorithms based on force-directed prin-
+ciples are normally used to embed graphs into planes in
+such a way that two connected (resp. disjoint) vertices are
+placed close to (resp. far from) each other, with a mini-
+mum (resp. maximum) distance between them (resp. from
+the plane’s center). In this context, in order to reflect in-
+herent symmetries, Fruchterman and Reingold [53] also pro-
+pose an efficient algorithm to place the vertices evenly in the
+plane, making the edges’ lengths uniform. For this purpose,
+each edge from the graph is treated as a spring that holds
+its endpoint vertices close to each other while a competing
+repulsive force is applied to push all vertices away from one
+another, even though they are not connected by an edge in
+the original graph. After enough iterations, the final sys-
+tem will reach equilibrium, minimizing then the difference
+between all attractive and repulsive forces.
+The repulsive and attractive forces fr and fa between two
+vertices are respectively given by equations (B1) and (B2).
+While k =
+�
+A/|V| is set to be related to the area A of the
+Euclidean plane, duv holds the distance between the vertices
+u, v ∈ V. Finally, the total energy ft of the system is given
+by adding the forces between all pairs of vertices, as shown
+in (B3). Therefore, ft goes to zero as the system approaches
+its equilibrium. This register embedding is hereafter referred
+to as spring layout.
+fr(u, v) = −k2/ruv
+(B1)
+fa(u, v) = d2
+uv/k
+(B2)
+ft =
+�
+u,v∈E
+fa(u, v) +
+�
+u∈V
+�
+v∈V:u̸=v
+fr(u, v)
+(B3)
+Fig. 10 depicts a possible embedding by directly applying
+the algorithm on a graph with 5 vertices and 7 edges. For a
+deep description of the algorithm, one may refer to [3, 53].
+If such a realization exists, the graph under consideration is
+naturally embedded as a UD graph if enough iterations are
+allowed (i.e., by iterating until the system reaches the equi-
+librium). It is also worthwhile mentioning that, as pointed
+out by [3], the proposed embedding strategy is only feasible
+on neutral atom-based QPUs once they respect the device’s
+technical constraints, such as maximum distance from the
+register’s center and minimum distance between atoms. If
+any technical constraint is violated, one might re-scaling ev-
+ery position vector by a factor α > 0.
+FIG. 10: Illustration of a register based on the spring layout
+for a given graph G = (V, E) with 5 vertices and 7 edges:
+the positions were generated with the Fruchterman-Reingold
+algorithm.
+Appendix C: Classical greedy pricing algorithm
+Algorithm 3 summarizes the proposed random weighted
+independent set generator. As input, the algorithm receives
+a graph G = (V, E, W), where W is the vertex weighting vec-
+tor. First, an auxiliary graph G′ is created to receive a copy
+of the input graph G (step 3). Then, a vertex from graph
+G′ is randomly selected and added to the independent set S
+(see steps 6-7). Next, the selected vertex and its neighbors
+are removed from G′ in step 8. Steps 6-8 are repeated until
+no vertex remains. The overall cost of the final solution is
+then calculated considering the vertex weighting vector W
+(see step 11).
+Due to the inherently stochastic nature of
+the proposed algorithm, different weighted independent sets
+might potentially be generated from the same input by run-
+ning steps 3-11 can be run multiple times (see the condition
+in step 2).
+Algorithm3 Random Independent Set Generator
+Input: A Graph G = (V, E), W, a threshold weight wmin, and
+the maximum number of tries t.
+Output: A set of weighted independent sets.
+1: S ← ∅
+2: while The maximum number of tries t is not reached do
+3:
+G′ ← G
+4:
+IS ← ∅
+5:
+while G′ has vertices do
+6:
+Randomly select a vertex in G′
+7:
+Add the selected vertex to IS
+8:
+Remove the selected vertex and its neighbors from G′.
+9:
+end while
+10:
+if IS was not generated before and its total weight is
+greater than wmin then
+11:
+S ← S ∪ IS
+12:
+end if
+13: end while
+14: return S
+
+3
+5
+2
+414
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@@ -0,0 +1,646 @@
+A unipolar quantum dot diode structure for advanced quantum light sources
+T. Strobel,1, ∗ J. H. Weber,1, ∗ M. Schmidt,2 L. Wagner,1 L. Engel,1 M.
+Jetter,1 A. D. Wieck,2 S. L. Portalupi,1 A. Ludwig,2 and P. Michler1
+1Institut für Halbleiteroptik und Funktionelle Grenzflächen,
+Center for Integrated Quantum Science and Technology (IQST) and SCoPE,
+University of Stuttgart, Allmandring 3, 70569 Stuttgart, Germany
+2Lehrstuhl für Angewandte Festkörperphysik,
+Ruhr-Universität Bochum, D-44780 Bochum, Germany
+(Dated: January 10, 2023)
+1
+arXiv:2301.03541v1  [quant-ph]  9 Jan 2023
+
+Abstract
+Triggered, indistinguishable, single photons play a central role in various quantum photonic implemen-
+tations. Here, we realize a novel n+−i−n++ diode structure embedding semiconductor quantum dots: the
+gated device enables spectral tuning of the transitions and deterministic control of the observed charged
+states. Blinking-free single-photon emission and high two-photon indistinguishability is observed. The
+linewidth’s temporal evolution is investigated for timescales spanning more than 6 orders of magnitude,
+combining photon-correlation Fourier spectroscopy, high-resolution photoluminescence spectroscopy, and
+two-photon interference (visibility of VTPI,2ns = (85.5±2.2) % and VTPI,9ns = (78.3±3.0) %). No spectral
+diffusion or decoherence on timescales above ∼ 9ns is observed for most of the dots, and the emitted pho-
+tons’ linewidth ((420±30) MHz) deviates from the Fourier-transform limit only by a factor of 1.68. Thus,
+for remote TPI experiments, visibilities above 74% are anticipated. The presence of n-doping only signifies
+higher available carrier mobility, making the presented device highly attractive for future development of
+high-speed tunable, high-performance quantum light sources.
+Quantum optical implementations and applications require sources of non-classical light ca-
+pable of emitting single photons on-demand and with a high degree of indistinguishability [1].
+Furthermore, for upscaling the experimental complexity, remote sources capable of emitting light
+at the same wavelength are highly desirable [2–6]. Semiconductor quantum dots have shown
+potentials to fulfill all the aforementioned needs, additionally being able to control the emission
+wavelength via various mechanisms, i.e. the application of electric [2] or magnetic field [7], as well
+as the use of external mechanical strain [8, 9]. Being embedded in a semiconductor matrix allows
+for the realization of high-quality photonic resonators [10–12], as well as the implementation of
+compact diode structures to control the emission wavelength via local tuning [13]. Interestingly,
+placing the quantum dots (QDs) into an electrically gated structure further provides a stabiliza-
+tion of the carrier environment which decreases the impact of spectral diffusion on the photon
+linewidth [14, 15]. These possibilities culminated in the realization of a source of single- and
+indistinguishable-photons with an end-to-end efficiency as high as 57% [16], having the sample
+been grown via molecular beam epitaxy (MBE) embedding the emitters into a p-i-n diode struc-
+ture. Despite these recent results, further improvements in the sample design and realization can
+be obtained: from the growth point of view, the presence of a p-layer in the implementation of
+the diode embedding the QDs requires the doping with carbon atoms which constitute an impurity
+2
+
+during growth as well as in the MBE chamber itself, even though there is a minimal memory ef-
+fect [15, 17, 18]. In high-frequency electrical device applications, using holes in p-doped structures
+instead of electrons in n-doped structures would limit the achievable speed due to their 20-times
+lower mobility allowing for fast control [19]. For example, a scheme theoretically proposed in
+Ref. [20] indeed requires electrical Stark shift of the QD transitions on sub-picosecond timescales
+to implement cavity-mediated single and photon-pair generation with QDs.
+For these reasons, in the present study, the quantum dots have been embedded into a n+−i−n++
+diode structure, while the light extraction is enhanced by a planar cavity design formed by two
+distributed Bragg reflectors (DBR) above and below the emitters. This novel diode structure
+was implemented performing a two-step growth combining metal organic vapor phase epitaxy
+(MOVPE) for the n-doped, bottom DBR, and MBE for the QD and gate diode, as well as the top
+DBR. On the one hand, MOVPE allows for a fast deposition of high-quality multilayers form-
+ing high-reflectivity DBRs, being the growth performed at high temperature and low vacuum.
+On the other hand, MBE with its high-vacuum and high purity growth conditions, enables the
+slow and controlled deposition of semiconductor nanostructures with a defect-free environment,
+resulting in high-coherence photon emission [6, 18, 21–23]. In the following, the optical and
+quantum optical properties of the emitted photons from the n+−i−n++ diode structure will be
+reported. In particular, high single-photon purity and indistinguishability have been observed re-
+spectively via Hanbury-Brown and Twiss, and Hong-Ou-Mandel-type experiments. Fabry-Perot
+interferometry allowed for measuring the static spectra of the emitted photons in high resolution
+and for comparing the linewidth with the expected Fourier-transform (FT) limit (obtained from
+decay time measurements). Photon-correlation Fourier spectroscopy (PCFS) [24, 25] proves that
+spectral broadening mechanisms are absent in the large majority of the investigated QDs, while
+small deviations from the FT limit act on timescales shorter than a few nanoseconds, in accordance
+with two-photon interference measurements. In line with pioneering p-i-n-type diodes the novel
+n+−i−n++ diode structure further enables the Stark tuning of the QD transitions, together with a
+precise control of the observed charge states in the dot. Finally, voltage dependent studies prove
+the stabilization effect of the diode structure, having a drastic reduction of the emission linewidth
+for the designed operation conditions.
+3
+
+Norm. RF Intensity
+0
+1
+Frequency detuning (GHz)
+−1
+0
+1
+data
+fit
+SR
+QD Layer
+AlAs
+GaAs
+GaAs n+
+GaAs n++
+AlGaAs n+
+Gold
+AlGaAs
+Norm. coincicences
+0
+1
+Time delay (ns)
+−10
+0
+10
+0
+1
+−1000
+0
+1000
+E-field
+Vg
+MOVPE MBE
+DBR
+DBR
+n
+n
+i
+a
+f
+e
+Wavelength (nm)
+904.5
+903.0
+901.5
+900.0
+Gate Voltage (mV)
+−2000
+−1000
+0
+T = 6K
+X0
+X-
+XX
+b
+Gate Voltage (mV)
+−600
+−300
+0
+c
+Norm. RF Intensity
+ 
+0
+14
+28
+42
+56
+70
+ 
+data
+fit
+√Excitation power (√nW)
+d
+Norm. Intensity
+0
+1
+FIG. 1. Sample description and characterization: a, Schematic of the sample structure. b, µ-PL of one
+selected quantum dot as a function of gate voltage using an AB pumping scheme. c, RF signal of the same
+QD, under resonant excitation of the trion transition. d, Resonant fluorescence intensity over the square root
+of the (pulsed) laser power. e, Second-order autocorrelation measurement of the investigated trion line: in
+the inset, long time delay data (with 13 ns binning) show the absence of blinking. The data are normalized
+to the Poissonian level. f, High-resolution FPI measurement of the trion linewidth (under pulsed resonant
+excitation for Vg = −540mV applied voltage). Data (solid blue), fit (solid red) and the system response
+(SR) function (dashed) are shown.
+DEVICE DESIGN AND RESONANT EXCITATION
+The sample employed has been grown by combining MOVPE and MBE techniques.
+A
+schematic view of its structure is depicted in Fig. 1a. In a first step, n+-doped bottom DBRs
+are grown with MOVPE. Then, the sample is removed from the reactor and shipped for the subse-
+quent MBE covering of the InAs QDs embedded in the n+−i−n++ gate diode (see Methods).
+We first investigate the micro-photoluminescence (µ-PL) spectrum (in above barrier (AB)
+pumping scheme at saturation) as a function of the applied gate voltage Vg. The acquired data
+4
+
+are depicted as a voltage/wavelength intensity map in Fig. 1b. Starting from a negative bias,
+several distinct lines appear and disappear as Vg is increased towards 0 V. Two-photon excitation
+(TPE) and decay time measurements (see Supplementary Fig. S1 and S3), allow for identifying
+the transitions as neutral exciton and biexciton, and negatively charged trion respectively (as this
+transition appears at a bias where a negative charged QD is energetically more favorable). At low
+pumping rates (see Supplementary Fig. S2), only trion or exciton lines are present for specific
+applied voltage values [26]: this controlled switching can be understood as a consequence of the
+Coulomb blockade. By applying gate voltages, the position of the discrete energy levels in the QD
+relative to the Fermi sea changes. Depending on the voltage level certain excitonic states can be
+favored.
+The emission lines undergo a wavelength shift (71.1±0.2 GHz V−1, ∼2.8·10−3 GHz per V cm−1)
+with changing Vg. This change in wavelength is the characteristic dc Stark shift [27] caused by
+the electrostatic field of the diode structure. These findings demonstrate the controlled switching
+of electronic states in this novel structure as well as the wavelength tunability with applied voltage.
+Fig. 1c shows the evolution of the trion wavelength under resonant pumping for various ap-
+plied voltages, where again a Stark shift of the emission wavelength can be observed (72.7 ±
+0.6 GHz V−1, ∼2.9 · 10−3 GHz per V cm−1). Under pulsed excitation, the trion resonance fluo-
+rescence intensity shows clear Rabi oscillations (see Fig. 1d), proving coherent control of the state
+population: the observed oscillations yield a state preparation fidelity of ≈ 85%.
+Under resonant excitation at the π-pulse, the second-order correlation plotted in Fig. 1e re-
+veals a pronounced anti-bunching at zero time delay, with a single-photon purity of g(2) (0) =
+0.028 ± 0.001. Furthermore no blinking is observed under resonant pumping in the whole time
+delays considered in the histogram of correlations (as shown in the inset of Fig. 1e with time sep-
+arations beyond ±1µs): this represents an important characteristic for the implementation of high
+brightness sources of quantum light. Besides that, the absence of p-doping helps in decreasing
+the presence of impurities in the QD host material, impurities which may also result in blinking
+dynamics at long time separation [28]. This blinking-free behavior is consistent with studies on
+similar diode-like heterostructures utilizing the Coulomb blockade to stabilize the charge envi-
+ronment [29]. For different applied voltages, it is also shown that resonant two-photon excitation
+(TPE) can be employed to generate exciton-biexciton pairs (see Supplementary Fig. S1).
+5
+
+Relative detuning (GHz)
+Norm. Intensiy
+0
+2
+4
+−4
+0
+4
+2
+-2
+a
+-470mV
+-660mV
+-630mV
+-540mV
+-450mV
+-440mV
+-420mV
+c
+b
+HOM I
+HOM II
+-420mV
+-450mV
+-540mV
+-660mV
+-440mV
+-470mV
+-630mV
+FT limit
+Linewidth (GHz)
+0
+1
+2
+3
+4
+Int. emission rate (arb. u.)
+0
+5
+10
+Gate voltage (mV)
+−750
+−600
+−450
+−300
+PCFS linewidth (GHz)
+0
+1
+2
+3
+4
+Time delay (s)
+1e−08
+1e−06
+1e−04
+1e−02
+FIG. 2. Linewidth investigation of the trion line at the π-pulse for different gate voltage levels: a,
+High resolution FPI spectra for different gate voltages (vertically shifted for clarity). b, Emission linewidth
+extracted from the FPI data in a (blue dots). For comparison the RF emission intensity extracted from the
+voltage map in Fig. 1c is shown in black. The two crosses indicate the conditions for the successive two-
+photon interference measurements of Fig. 3 and Tab. I. c, PCFS measurements from ns to ms timescales.
+The estimated lifetime-limited linewidth is shown in black (FL). For each voltage the linewidth is a straight
+line indicating no spectral dynamics on the mentioned timescales. Moving away from the center position of
+the charge plateau in the voltage scan an increase of the emission linewidth is observed.
+A charge-stable QD environment, in combination with a low density of impurities and defects,
+is also supposed to positively impact the emission linewidth. For this reason, we investigate the
+emission linewidth under resonant pumping at the π-pulse. First, a scanning Fabry-Perot inter-
+ferometer (FPI) is employed for recording high-resolution stationary spectra of the investigated
+QDs (see Methods). Fig. 1f shows the measured emitted photons’ linewidth for the trion transition
+under investigation reaching a value of (420±30) MHz, broadened only by a factor of 1.68 com-
+pared to the FT limit (∆νFL = (250±20) MHz). This measured close-to lifetime-limited linewidth
+demonstrates a modest impact of dephasing and spectral diffusion in the presented heterostructure.
+LINEWIDTH TEMPORAL EVOLUTION OVER APPLIED GATE VOLTAGE
+Following this first characterization, we conduct FPI measurements for various applied gate
+voltages, and the results are summarized in Fig. 2a. As it can be seen, the observed linewidth
+6
+
+varies drastically with the applied bias. The emission linewidth extracted from fitting the FPI
+measurements is reported in Fig. 2b (blue dots): interestingly, a voltage range exists for which
+the linewidth reaches minimal values which corresponds to a maximum in the observed µ-PL in-
+tensity (black solid line). Outside this plateau, the measured linewidth increases, accompanied
+by a decrease in luminescence. In line with previous studies, the observed line broadening can
+be attributed to cotunneling [30–33]: the QD is tunnel-coupled to the Fermi sea, with increased
+tunneling interaction for voltage levels outside the shown stable region, hence decreasing the pho-
+ton coherence. At the center of the plateau tunnel coupling is inhibited increasing the photon
+coherence time, hence minimizing the linewidth.
+Despite the narrow linewidth observed at the center of the voltage plateau, it is useful to investi-
+gate the origin of potential dephasing and spectral diffusion mechanisms responsible for the small
+deviation from the Fourier limit. To do so, PCFS is employed [24, 25]. PCFS enables measuring
+the time evolution of the linewidth of the emitted photons, with high temporal and spectral resolu-
+tion and for timescales that can vary from few nanoseconds to few milliseconds (see Methods).
+Exemplary results of PCFS measurements are reported in Fig. 2c, where the applied gate voltage
+has also been varied in order to provide a profound insight on the impact of the applied electric
+field on the emission linewidth. As it can be seen, the PCFS results show that, for all applied
+voltages, the linewidth remains constant over time, where only the absolute value is changing.
+The lowest measured linewidth in PCFS is 520 ± 100MHz (Vg = −570mV) which is very close
+to the FT limit (dashed line) as it also was observed in the previous FPI measurement recorded for
+similar applied voltages (see Fig. 1f). The PCFS measured linewidth is close to the resolution limit
+of the PCFS setup, which could explain the slightly higher value with respect to the analogous FPI
+value, which corresponds to the stationary limit of the emission spectrum [34]. This agreement
+between PCFS at long timescales and FPI measurements indicate that any deviation from the FT
+limit is due to dephasing and spectral diffusion mechanisms happening at timescales shorter than
+10 ns. The same conclusion applies for the other applied voltages, since the long timescale PCFS
+values match the corresponding FPI measurements (see Fig. 2b and Supplementary Fig. S4). This
+supports the conclusion that the process responsible for this additional line broadening happens
+at a timescale shorter than 10ns, since there is no further broadening, even for times longer than
+10ms.
+The discussed results, further supported by previous studies [14, 30, 33], clearly indicate the
+existence of a voltage range for which the linewidth is minimized. There, the small deviation from
+7
+
+FT limit is proven to be due to mechanisms with dynamics faster than 10ns.
+SHORT TIMESCALE DYNAMICS VIA VOLTAGE-DEPENDENT TWO-PHOTON INTERFER-
+ENCE
+To probe even shorter timescales, two-photon inference (TPI) measurements with consecutively
+emitted photons are conducted (time separation between 2ns and 9ns), therefore extending the
+investigated time range close to the radiative lifetime of the emitter. The measurements in Fig. 3a
+show the central peaks around zero time delay with a clear signature of two-photon interference,
+for an applied gate voltage within the previously observed plateau (here Vg = −570mV). In this
+configuration, a TPI visibility as high as V −570mV
+TPI
+= (85.8±2.2) % is recorded for a photon time
+separation of 2ns. This implies that the mechanism responsible for the small deviation from near-
+unity visibility happens at timescales shorter than 2ns. Considering the rather small size of the
+QDs, the effective coupling to acoustic phonons [35] could result in a dephasing mechanism which
+acts at below nanosecond timescales: this would explain the deviation from FT limit and the re-
+spective impact on the TPI.
+To further investigate the broadening mechanisms, TPI measurements with time separations of 4ns
+and 9ns were performed. An overview of the results is given in Tab. I. For the center of the charge
+plateau at Vg = −570mV a decrease from VTPI,2ns = (85.5±2.2) % to VTPI,4ns = (79.7±2.5) %
+(for 4ns time separation) and to VTPI,9ns = 78.3±3.0% (for 9ns) in the TPI visibility is observed.
+This rather modest decrease of the TPI visibility can be attributed to spectral broadening mech-
+anisms happening at such timescales (still not resolvable in PCFS measurements). Intriguingly,
+estimating the reachable TPI visibility from the recorded stationary linewidth in FPI measurements
+(Fig. 2a), a value of V sim
+TPI = (74.3±1.5) % is expected [36]. This quantity would also represent the
+achievable TPI visibility for photons stemming from remote sources. This value matches, within
+margin of uncertainty, the observed value of VTPI,9ns = (78.3±3.0) %. This indicates that, to-
+gether with broadening mechanisms happening on timescales smaller than 2ns, a small linewidth
+broadening is further observed between 2ns and 9ns. Above this timescale value, no further broad-
+ening mechanisms are observed: this is confirmed independently by PCFS measurements, and by
+the agreement between the observed TPI at 9ns and the expected two-photon interference inferred
+from the stationary spectrum (measured with FPI measurements).
+Interestingly, for voltages outside the discussed stable region (here Vg = −450mV), the
+8
+
+Vg(mV) VTPI,2 ns(%) VTPI,4 ns(%) VTPI,9 ns(%) Vsim
+TPI(%)
+−570
+85.5±2.2
+79.7±2.5
+78.3±3.0 74.3±1.5
+−450
+14.9±3.6
+13.9±3.7
+14.6±3.5 14.5±1.0
+TABLE I. TPI visibilities for different photon temporal delays and applied gate voltages. Moreover the
+expected TPI visibility (Vsim
+TPI) of statistically independent QD emissions is extracted from the static FPI
+measurements.
+observed TPI is drastically reduced.
+As shown in Fig. 3b, a value as low as V −450mV
+TPI
+=
+(14.9±3.6) % is observed. The applied voltage has been chosen in a range which has already a
+clear impact on the emission linewidth, despite a modest reduction of the observed count rate (see
+Fig. 2b). These findings are consistent with the observation on the line broadening: while for the
+voltage range within the plateau the electric field stabilizes the environment, resulting in a close
+to FT limited linewidth, outside this charge plateau the situation changes. There, carrier tunneling
+induces a broadening of the linewidth, and the tunneling itself happens on time scales shorter than
+10ns (as seen in PCFS data) and than 2ns (as seen in TPI measurements). More than 95 % of the
+investigated quantum dots (20 QDs) showed a linewidth behavior comparable with the reported
+results (As depicted in Supplementary Fig. S5, only one dot showed linewidth temporal dynam-
+ics in PCFS data, compatible with the presence of a local carrier trap next to the dot). Outside
+this stable plateau, the TPI visibility remains low (≈ 14%) for all photon time separations and it
+matches the value inferred from the linewidth: once again, this confirms the impact of dephasing
+on timescales shorter than 2ns (see results in Table I).
+CONCLUSIONS
+In this study, MOVPE and MBE have been combined to realize high optical quality quantum
+dots, embedded into a novel n+−i−n++ diode, and within a planar cavity formed by two DBRs.
+The high quality DBR is ensured by the high temperature (and high speed) deposition of MOVPE,
+while the MBE ensures the growth of QDs within low defect material structure. This, in com-
+bination with the embedding diode enables the observation of close-to-FT limit QD linewidth
+((420±30) MHz), where high coherent control of the population is observed as well as high
+single-photon purity (g(2) (0) = 0.028±0.001). The sample is investigated combining PCFS, FPI
+9
+
+0
+1
+Delay time (ns)
+−4
+0
+4
+Norm. coincidences
+0
+1
+b
+a
+HOM I: Vg = -570 mV
+HOM II: Vg = -450 mV
+fit
+data
+fit
+FIG. 3. Two-photon interference: TPI measurement from the trion at the π-pulse for two different gate
+voltages. Double pulses with 2 ns temporal separation were used to create the interfering photons. Un-
+wanted laser background (or broad tails from phonon assisted emission) is filtered via a transmission spec-
+trometer of spectral width ∆filter = 15GHz, much larger than the emission linewidth. The blue areas show
+the measured data for parallel polarization, with their fit function in red. The brown areas correspond to
+the fitted data of the orthogonal polarization measurement. a, TPI results for a voltage of Vg = −570mV
+corresponding to the center of the charge plateau. b, measurement results for a voltage of Vg = −450mV at
+the edge of the charge plateau (compare with Fig. 2b).
+and TPI measurements enabling a study on emission linewidth and dynamics of the decoherence
+mechanisms for timescales spanning more than 6 orders of magnitude. In particular, the small de-
+viation from lifetime-limited spectra is investigated via PCFS and FPI measurements: combining
+these two techniques it is possible to conclude that no spectral broadening mechanisms happen for
+timescales between 10ns and the stationary limit (set by the FPI results). Furthermore, it has been
+10
+
+clearly proven that the applied gate voltage allows for wavelength tuning and stabilizing the QD
+charge environment, within a defined voltage range where the luminescence is maximized. Out-
+side from this ideal plateau, the emission linewidth increases and the brightness decreases. Still,
+thanks to PCFS we can conclude that the mechanisms responsible for this degradation outside the
+ideal voltage range have a timescale below 10ns. Finally, two-photon interference measurements
+are employed to investigate the timescales between 2ns and 9ns to provide further insights in a
+time range not accessible by PCFS. These measurements show that, even for the smallest time
+separation, the linewidth is still slightly impacted by fast dephasing mechanisms (below 2ns),
+which we attribute to acoustic phonon coupling. Still, a small degradation of the TPI visibility is
+observed increasing the photon time separation to 9ns: in this condition, the measured visibility is
+consistent with the one expected by simulations, taking into account the stationary linewidth mea-
+sured via FPI. This further demonstrates that the only decoherence mechanisms, despite modest,
+are happening at relatively short timescales.
+Phonon coupling can be reduced in the future by growing larger QDs (which couple less
+to phonons) or by integrating the emitters in photonic microcavities: Purcell enhancement is
+known to be beneficial for decreasing the impact of dephasing mechanisms, in particular at short
+timescales, further improving the photon indistinguishability. Both approaches, i.e. growth of
+larger QDs and use of photonic cavities, are fully compatible with the realized sample design. In-
+terestingly, the absence of p-doping will enable in the near future the realization of devices where a
+fast AC bias can be applied, thanks to the high electron mobility. Finally, the observed Stark shift
+induced by the diode structure will have important application in realizing wavelength tunable
+sources, fundamental requirement for the implementation of TPI from distinct devices: already
+at the present stage, a remote visibility of ≈ 74% would be expected, a value much higher than
+other reported for InGaAs QDs, only exceeded recently by GaAs quantum dots [6]. This makes
+the implementation of an n+−i−n++ diode, embedding (larger) QDs into optical microcavities
+highly desired for future quantum photonics.
+Acknowledgments
+The authors gratefully acknowledge the funding by the German Federal Ministry of Education
+and Research (BMBF) via the project QR.X (No. 16KISQ013) and the company Quantum Design
+for their persistent support. We would like to thank Sergej Vollmer for the support in MOVPE
+11
+
+growth.
+Author contributions
+T.S., J.H.W. and L.W. performed the measurements and analyzed the data. M.S. and A.L. grew
+the sample with the support of A.D.W.. L.E. fabricated the device. A.L. and M.J. designed the
+sample. T.S. and S.L.P. wrote the manuscript with the support of J.H.W and P.M.. A.L., M.J.,
+S.L.P., A.D.W. and P.M. coordinated the project. All authors contributed to scientific discussions
+and revision of the manuscript.
+∗ These authors contributed equally
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+METHODS
+Sample design. The sample employed has been grown by combining MOVPE and MBE
+growth. A schematic view of the sample structure is depicted in Fig. 1a. Starting with MOVPE,
+the heterostructure was grown on a (100)-oriented n+-doped GaAs substrate, starting with a
+15
+
+300 nm thick n+-doped GaAs layer followed by 29 n+-doped pairs of Al0.95Ga0.05As (77 nm) and
+GaAs (64 nm) forming the bottom DBR. The MOVPE growth is completed with a last layer of
+Al0.95Ga0.05As and a thinner GaAs (31 nm) layer, both n+-doped. From this point on the sample is
+shipped in atmospheric conditions to continue the growth with MBE. After careful oxide removal,
+the gate contact is formed by a 28 nm thick GaAs (n+, 2×1018 cm−3) layer and then followed by
+one (additional) GaAs/AlAs DBR pair. 134.5 nm thick Al0.34Ga0.66As functions as the current
+blocking layer and enables an electrostatic potential across the optically active region. After 5 nm
+of undoped GaAs functioning as spacer, the electron wetting layer state-free self-assembled InAs
+QDs [37] are grown in the Stranski-Krastanow mode with a subsequent flushing step. The result-
+ing QD ensemble luminescence is peaked around 910 nm. Tunnel coupling of the QDs with the
+Fermi sea is realized by a succeeding 35 nm thick GaAs tunnel barrier. In order to reach the back
+contact in a later step, an etch stop layer is added consisting of 1 nm GaAs followed by 1.5 nm
+AlAs and 41 nm GaAs in growth direction. To further increase the extraction of photons, four
+more undoped GaAs/AlAs DBRs are deposited. The sample is completed by an 78nm etch stop
+layer follwoed by the 64,25nm tick GaAs cap.
+The utilized device shows a QD density of 0.1 − 1 · 106 cm−2 of QDs emitting in the wavelength
+range between 899 nm and 911 nm as measured by µ-PL maps. After metallizing the highly
+doped substrate with a chromium-gold alloy, silver epoxy glue is used to contact the gate to the
+chip carrier. In order to apply a voltage to the n++- doped top layer, UV photolithography and
+wet-chemical etching are used to provide access to the back contact, leaving unaffected the DBRs
+on top of the investigated QDs. Alternating wet-chemical etching using citric acid with hydrogen
+peroxide (H2O2) is employed to remove the GaAs layers, and hydrochloric acid (HCl) deluted
+with deionized water to remove the AlAs layers. For contacting the back contact a titanium-
+platinum-gold alloy is deposited via electron-beam physical vapor deposition and subsequently
+wire-bonded with a 150 µm diameter In wire.
+Experimental configuration. Above barrier (AB) and resonant fluorescence (RF) pumping
+are achieved via a continuous wave diode laser and a Ti:sapphire pulsed laser system respectively.
+The sample is placed inside a closed-cycle He cryostat (Montana Instruments, Cryostation s50)
+and kept at a temperature of T = 6K. The excitation and light extraction is accomplished via
+a confocal microscopy setup, with the microscope objective located inside the cryostat. For RF
+detection, a cross-polarization setup is employed (suppression ∼ 107) [38]. The signal is filtered
+16
+
+utilizing a monochromator based on a transmission grating with a spectral resolution of ∼ 15GHz.
+PCFS and TPI measurements were carried out with high efficiency (ηSPAD = 30%) single-photon
+avalanche diodes (SPAD) with a temporal resolution of ∆τSPAD = 350ps. The detectors utilized in
+the lifetime measurements had a higher temporal resolution (∆τSPAD = 50ps) at a cost of reduced
+efficiency (ηSPAD = 2%).
+Ti-sapphire laser. For the pulsed resonant excitation of the QDs, a Coherent Mira Ti:sapphire
+laser was utilized. The laser creates pulses of ≈ 3ps with a repetition rate of 76.2MHz. For the
+PCFS measurements, this rate is amplified (×4) to 304.8MHz.
+Fabry-Perot interferometer and analysis.
+The FPI utilized in the measurements has a
+free spectral range of 15 GHz and a resolution of 100MHz (extracted from the width of the
+SR function). The data in Fig. 1f were fit with a Voigt function convolved with the measured
+SR function. The Voigt profile includes homogeneous and inhomogeneous contributions: with
+the reasonable assumption that the homogeneous linewidth can be extracted from the decay
+time measurement of the trion line (τdec = (652±5) ps, yielding a homogeneous linewidth of
+∆νhomo = (250±20) MHz), this contribution is kept fixed, while the inhomogeneous broadening
+is left as free fitting parameter.
+Photon-correlation Fourier spectroscopy. PCFS employs a Mach-Zehnder interferometers
+where both exits of the beamsplitter are collected, measured by single-photon counting modules
+and time correlated. In our experiment the translation stage was moved up to 372 mm path dif-
+ference, whereas every 4 mm path difference a correlation measurement was carried out with a
+scanning rate of ∼ 10fringess−1. The consequential frequency resolution in the spectral correla-
+tion is 806 MHz. As this emitter shows close-to lifetime-limited linewidth, a Lorentzian lineshape
+is assumed for the analysis. Under this assumption, the frequency resolution for the measured
+spectrum is increased by a factor of two to 403 MHz. The maximum spectral range covered in
+this framework is 37.5 GHz. The excitation rate of 304.8MHz would allow a temporal resolution
+of ∼ 3ns. However, the current temporal resolution is limited to ∼ 10ns, as the required small
+bin width at these short timescales drastically inflates the statistical error. At timescales reaching
+the scanning rate of the translation stage artifacts of the selfsame distort the signal. The temporal
+upper bound is therefore set to ∼ 10ms.
+17
+
+Two-photon interference. The experiment is performed with an unbalanced Mach-Zehnder
+interferometer (MZI) operating with variable time separations of τMZI = 2, 4 and 9ns.
+18
+

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@@ -0,0 +1,1582 @@
+Learning Trajectory-Word Alignments for Video-Language Tasks
+Xu Yang1
+Zhangzikang Li1,2
+Haiyang Xu2*
+Hanwang Zhang3
+Qinghao Ye2
+Chenliang Li2
+Ming Yan2
+Yu Zhang1*
+Fei Huang2
+Songfang Huang2
+1Southeast University
+2Alibaba Group
+3Nanyang Technological University
+{xuyang palm, zhang yu}@seu.edu.cn, lizhangzikang@gmail.com, {shuofeng.xhy, yeqinghao.yqh,
+lcl193798, ym119608, f.huang, songfang.hsf}@alibaba-inc.com, hanwangzhang@ntu.edu.sg
+Abstract
+Aligning objects with words plays a critical role in
+Image-Language BERT (IL-BERT) and Video-Language
+BERT (VDL-BERT). Different from the image case where
+an object covers some spatial patches, an object in a video
+usually appears as an object trajectory, i.e., it spans over
+a few spatial but longer temporal patches and thus con-
+tains abundant spatiotemporal contexts.
+However, mod-
+ern VDL-BERTs neglect this trajectory characteristic that
+they usually follow IL-BERTs to deploy the patch-to-word
+(P2W) attention while such attention may over-exploit triv-
+ial spatial contexts and neglect significant temporal con-
+texts.
+To amend this, we propose a novel TW-BERT
+to learn Trajectory-Word alignment for solving video-
+language tasks. Such alignment is learned by a newly de-
+signed trajectory-to-word (T2W) attention. Besides T2W at-
+tention, we also follow previous VDL-BERTs to set a word-
+to-patch (W2P) attention in the cross-modal encoder. Since
+T2W and W2P attentions have diverse structures, our cross-
+modal encoder is asymmetric. To further help this asymmet-
+ric cross-modal encoder build robust vision-language as-
+sociations, we propose a fine-grained “align-before-fuse”
+strategy to pull close the embedding spaces calculated by
+the video and text encoders. By the proposed strategy and
+T2W attention, our TW-BERT achieves SOTA performances
+on text-to-video retrieval tasks, and comparable perfor-
+mances on video question answering tasks with some VDL-
+BERTs trained on much more data. The code will be avail-
+able in the supplementary material.
+1. Introduction
+By witnessing the boom of BERT-like models in single
+domains, e.g., vision or language [20, 44, 51], researchers
+begin to build vision-language BERTs [3, 9, 61] for learn-
+ing robust cross-modal associations. Compared with the
+*Corresponding authors.
+A boy is playing basketball.
+(a) RoI based IL-BERT
+(b) Patch based IL-BERT
+(c) Patch based VDL-BERT
+(d) TW-BERT
+Figure 1. The comparisons of four different ways to build object-
+word alignments. In (b), by patch-to-word (P2W) attention, im-
+plicitly object-word alignment can be built, e.g., the “ball region”
+is aligned with basketball. While in (c), P2W attention may con-
+centrate the attention of an object on only one frame, e.g., basket-
+ball only attends over on the “ball region” of the first frame. (d)
+By TW-BERT, the trajectory of an object is constructed, e.g., the
+“ball trajectory” through 4 frames is built.
+images, videos provide more details for building robust as-
+sociations, e.g., the spatiotemporal contexts in videos can
+better describe the actions. However, directly addressing
+dynamic videos may cost more storage and computation re-
+sources. To circumvent such huge cost, researchers first
+study image-language BERT (IL-BERT) [7, 16, 24, 26, 61]
+and then exploit the research fruits to build the challeng-
+ing, while more pragmatic, video-language BERT (VDL-
+BERT) [2,12,25,32–34,46,47,53].
+1
+arXiv:2301.01953v1  [cs.CV]  5 Jan 2023
+
+Pioneering techniques are directly inherited from IL-
+BERT into VDL-BERT, e.g., the image encoders are used
+to embed sampled video frames [22, 62]. However, since
+image encoders can hardly learn temporal contexts, the re-
+sultant VDL-BERTs will degrade to IL-BERT even if they
+are trained by video-text pairs. To ameliorate it, researchers
+apply the video encoders [4,11,23,29] to embed spatiotem-
+poral contexts. Though promising improvements are ob-
+served, these VDL-BERTs still neglect or ill-consider a
+significant factor in IL-BERT: the object-word alignment,
+which helps learn robust associations.
+Aligning the visual objects with the semantic knowledge
+lays the foundation of humans’ visual reasoning and so does
+the AI agent.
+To learn object-word alignments, as Fig-
+ure 1(a) shows, researchers use an RoI-based extractor to
+embed an image into a series of RoI features and train the
+model by predicting the masked RoI categories based on
+the paired texts [7,26,31]. However, this RoI-based extrac-
+tor is offline trained by object detection with limited label
+inventory, which will weaken the IL-BERT since the ex-
+tractor will not be updated during large-scale pre-training.
+To enable the end-to-end training, researchers substitute the
+RoI-based extractor with visual transformers whose outputs
+are a series of grid embeddings, which will be used to build
+the cross-modal connections. Although a single grid usu-
+ally does not construct an integral object, fortunately, the
+widely applied patch-to-word (P2W) attention of IL-BERT
+can softly seek the salient visual regions for a given query
+word. Then the object-word alignments can still be built
+between this softly detected object and the word, e.g., as
+shown in Figure 1(b), the object “boy” and “basketball” can
+be implicitly attended by the corresponding word queries.
+Although the P2W attention remedies the loss of RoI-
+level features for learning object-word alignments in IL-
+BERT, its effectiveness is weakened in the video case. This
+is because the objects usually act as the Trajectories which
+span a few spatial while multiple temporal grids in the
+videos.
+Thus, directly applying the P2W attention may
+over-exploit the trivial spatial contexts while neglecting the
+significant temporal contexts and then make the model at-
+tend to only one or two frames.
+Figure 1(c) shows this
+limitation that P2W attention only aligns the “ball” in the
+first frame to the word ball. To address this limitation, we
+propose to build Trajectory-to-Word alignments to solve
+video-language tasks and name this model as TW-BERT.
+Specifically, such alignment is learnt by a novel designed
+trajectory-to-word T2W attention, which first uses the word
+as the query to seek the salient parts of each frame and the
+sought parts are sequenced to form the trajectory. Then the
+query word attends over the trajectories again for captur-
+ing cross-modal associations. In this way, the trivial spa-
+tial regions are weakened and the temporal contexts will be
+strengthened, e.g., as shown in Figure 1(d), the attention
+weights of the word will be concentrated on the object tra-
+jectory instead of only on one or two frames as in (c).
+For implementation, we follow most VDL-BERTs to set
+up the network: two single-modal encoders for the video
+and text and one cross-modal encoder, which is sketched
+in Figure 2.
+For the cross-modal encoder, we also ap-
+ply a word-to-patch (W2P) attention as previous VDL-
+BERTs [2, 22, 23, 47] to assign a visual patch with certain
+words. Since our T2W attention operates a double atten-
+tion mechanism, which does not have the same structure
+as W2P, our cross-modal encoder is asymmetric. Further-
+more, to help this asymmetric encoder build more robust
+trajectory-word alignments, we apply the align-before-fuse
+strategy [24].
+This strategy learns to pull close the em-
+beddings of paired video-text data output from two single-
+modal encoders. However, different from [24] which uses
+global [CLS] tokens to calculate the coarse-grained con-
+trastive loss, we propose a fine-grained one by summing the
+token-wise maximum similarities between the embeddings
+of the patches and words. In this way, we amplify the influ-
+ences of the local details to avoid the single-modal encoders
+over-exploiting the trivial global background contexts. To
+sum up, our contributions are:
+• We propose a novel perspective to consider the videos
+that the videos are composed of moving object trajec-
+tories, which may inspire the researchers to build more
+advanced VDL-BERTs.
+• We propose a simple while effective T2W attention
+to learn Trajectory-to-Word alignments that extends
+the object-word alignment in the image case, which is
+achieved by patch-word alignment, to the video case.
+• We propose the fine-grained contrastive loss in VDL-
+BERT to exploit more local details to pull close the
+video and word embeddings.
+• We achieve SOTA scores under the condition of using
+the same amount of training data.
+2. Related Work
+2.1. Image-Language BERT (IL-BERT)
+Recently, various techniques have been proposed in IL-
+BERT to learn vision-language connections. Most of them
+aim at capturing robust object-word alignments since such
+alignments construct the foundations of visual reasoning.
+In the beginning, a pre-trained Faster-RCNN [41] is used
+to extract a series of RoI poolings, where each one con-
+tains one or a few salient objects, to facilitate building
+object-word alignments [7,26,31,48]. However, this Faster-
+RCNN is usually pre-trained by the object annotations from
+COCO [6] and VG [19], whose concept space is much
+narrower than the data used to train IL-BERT, which can
+be almost unlimitedly collected from the websites. More-
+over, this Faster-RCNN is not updated during the end-to-
+end training of the IL-BERT, which means that the visual
+2
+
+encoder may hardly learn novel knowledge from the web-
+collected data. Thus the performances of these IL-BERTs
+are limited.
+To further release the potential of hugely web-collected
+data, the offline Faster RCNN is switched into vision Trans-
+formers [10, 29] and thus a homogeneous IL-BERT, i.e.,
+all the components are transformer-based, is built, which
+is more easily trained end-to-end [31]. Compared with the
+RoI-based encoder, the vision Transformer outputs a series
+of patch embeddings and thus may lose object-level knowl-
+edge. To remedy such loss, various strategies are proposed
+to improve the vision-language alignments. For example,
+the align-before-fuse strategy [24] aligns the paired image-
+text embeddings before cross-modal fusion for facilitating
+the subsequent fusion.
+And the fine-grained contrastive
+objective [58] amplifies the local details for learning the
+object-word alignments. Motivated by them, in our VDL-
+BERT, we propose a fine-grained align-before-fuse strat-
+egy to make the later asymmetric cross-modal encoder learn
+more robust word-trajectory alignments.
+2.2. Video-Language BERT (VDL-BERT)
+Since the spatiotemporal contexts in videos can hardly
+be learnt by image extractors, only inheriting the techniques
+which are successful in IL-BERT to VDL-BERT is not
+enough. Thus, based on the fruits of IL-BERT, most VDL-
+BERTs aim at exploiting more spatiotemporal contexts to
+build cross-modal associations. One straightforward way is
+to learn such spatiotemporal contexts through video Trans-
+formers [4, 27]. Besides this, some more advanced tech-
+niques are proposed, e.g., VIOLET [11] tokenizes the dy-
+namic video patches and predicts the labels of these tokens;
+BridgeFormer [13] erases the words (nouns or verbs) from
+the text and learn to match the visual embeddings queried
+by the erased words and the remained texts; or ALPRO [23]
+computes the similarities between the video embeddings
+with the generated entity prompts. Although substantial im-
+provements are observed, these methods use video patches
+in the cross-modal encoder, which neglects that an object
+usually acts as the trajectory in the videos, and thus they
+may over-exploit the trivial spatial contexts. To ameliorate
+this limitation, we propose the Trajectory-Word attention to
+learn more robust vision-language alignment.
+3. Approach
+Figure 2 sketches TW-BERT, which has two single-
+modal encoders for embedding the video and text (cf.
+Sec. 3.1) and one cross-modal encoder for learning video-
+language associations (cf. Sec. 3.2). Different from the pre-
+vious VDL-BERTs, our cross-modal encoder is an asym-
+metric one that contains a classic word-to-patch (W2P) at-
+tention and a novel proposed trajectory-to-word (T2W) at-
+tention (cf.
+Sec. 3.2) for learning trajectory-word align-
+Text 
+Encoder
+Video 
+Encoder
+[CLS] 
+A
+boy
+is
+playing 
+basketball
+Video 
+Encoder
+Video 
+Encoder
+Video 
+Encoder
+Lc
+Trajectory-to-Word
+Attention
+Word-to-Patch
+Attention
+…
+…
+…
+…
+…
+Asymmetric
+Cross-modal
+Encoder
+Lf
+Lvtm
+Lmlm
+[CLS]
+[CLS]
+Figure 2. The architecture of TW-BET, which contains two single-
+modal encoders and one asymmetric cross-modal encoder. Totally
+four losses in the green blocks are used to train the whole model:
+coarse-/ fine-grained contrastive losses Lc/Lf, Lmlm, and Lvtm.
+ments. To further encourage the cross-modal encoder to
+learn such alignments, we propose a fine-grained align-
+before-fuse strategy which amplifies the local details to help
+single-modal encoders pull close the embeddings of paired
+video-texts (cf. Sec. 3.3).
+3.1. Single-Modal Encoders
+Video Encoder.
+For a video, we sample 4 224 × 224
+frames and input them into the 12-layer TimeSformer [4,23]
+for embedding.
+TimeSformer first partitions each frame
+into 14 × 14 non-overlapping patches, which are flattened
+and fed to a linear projection layer to produce a sequence
+of patch tokens. Then TimeSformer applies self-attention
+along the temporal and spatial dimensions to calculate per-
+frame features.
+These features are further mean-pooled
+along the height and width dimensions. Learnable spatial
+positional embeddings are added to each video token in the
+same spatial location of different frames. The final out-
+put embedding set is V
+= {vcls, v1, ..., vNV −1}, where
+vn ∈ Rd and vcls is the global [CLS] embedding.
+Text Encoder. For a text, we use a 6-layer transformer to
+embed it and the output is X = {xcls, x1, ..., xNX−1},
+where xn ∈ Rd and xcls is the global [CLS] embedding.
+Similar to the video encoder, we also add positional embed-
+dings to the text tokens.
+3.2. Asymmetric Cross-Modal Encoder
+After embedding videos and texts, a cross-modal en-
+coder is used to fuse them by calculating bi-directional as-
+sociations: vision-to-language and language-to-vision. No
+matter what the direction is, the motivation is to assign
+semantic knowledge from one domain to another. Since
+a single word contains integral semantic knowledge, we
+follow previous VDL-BERT [2, 22, 23, 47] to set a tradi-
+tional word-to-patch (W2P) attention to assign the words to
+a patch. However, different from the word, only the object
+instead of one grid conveys integral semantic knowledge.
+3
+
+Time
+Time
+(a): Patch-to-Word Attention
+(b): Trajectory-to-Word Attention
+Figure 3. The comparisons between Patch-to-Word (P2W) and
+Trajectory-to-Word (T2W) attentions, where the green block de-
+notes a query word and the blue blocks denote the video patches.
+In P2W attention, the word attends over all the video patches and
+may only concentrate on one frame, while in T2W attention, the
+salient parts of each frame are found to construct a trajectory,
+which is connected by the green line.
+In videos, an object usually spans both spatial and tempo-
+ral axes and thus the previously used patch-to-word (P2W)
+attention may fail to transfer the semantic knowledge. To
+amend this, we design a novel Trajectory-to-Word (T2W)
+attention for transferring semantic knowledge from videos
+to texts. Since W2P and T2W attentions have diverse struc-
+tures, our cross-encoder is asymmetric.
+Specifically, both W2P and T2W attentions are built on
+the Multi-Head Attention (MHA) operation, here we first
+formalize MHA and then introduce how to use it to build
+W2P and T2W attentions. Formally, MHA is*:
+Input:
+Q, K, V
+Att:
+Ai = Softmax(QWQ
+i (KWK
+i )T
+√
+d
+)
+Head :
+Hi = AiVWV
+i ,
+Multi-Head:
+H = [H1, H2, ..., Hh]WH,
+Output:
+Z = LN(H + Q),
+(1)
+where WQ
+i , WK
+i , WV
+i , WH
+i
+are all trainable matrices; h is
+the head number and dh = d/h; Ai is the i-th attention
+matrix corresponding to the i-th head matrix; [·] is the con-
+catenation operation; and LN is the Layer Normalization.
+Word-to-Patch (W2P) Attention. To calculate the W2P
+alignment, we apply the conventional W2P attention [23]:
+ZW 2P = MHA(Q = V , K = V = X),
+(2)
+where V ∈ RNV ×D, X ∈ RNX×D are respectively video
+and word embedding sets got from two single-modal en-
+coders. By setting the query Q to the video patch embed-
+dings, Eq. (2) learns to assign suitable words to each video
+patch and thus captures the W2P alignment.
+Trajectory-to-Word (T2W) Attention. We propose the
+T2W attention that uses two steps to learn the T2W align-
+ment: it first constructs a trajectory for a given word and
+then uses the word as the query to attend over the trajec-
+tory for capturing the associations. For convenience, we
+*To avoid symbol confusions, we use the calligraphic font (\mathcal
+commend in LaTex) to denote the built-in variables of the MHA module.
+Fine-Grained Word Patch Assignments
+Figure 4. For each word token xm, a corresponding video patch
+v∗
+m that has the maximum inner product with xi is found, which
+is indexed by the same color.
+introduce how the T2W attention calculates the fusion em-
+bedding zT 2W for a single word x and it is straightforward
+to extend it to a sequence of the words.
+In the first step, T2W attention uses x as the query to
+find the salient parts for each frame and then sequence these
+parts to construct the trajectory. Assuming Vt is the embed-
+ding set of the t-th frame, the salient part yt is got as:
+yt = MHA(Q = x, K = V = Vt).
+(3)
+Then the salient parts at different time frames construct a
+continuous flow Y = {y1, ..., yT }, which is the trajectory
+of the given word.
+In the second step, to get the trajectory-to-word fusion
+embedding zT 2W , we treat x as the query again while using
+the trajectory Y as the key and value in MHA:
+zT 2W = MHA(Q = x, K = V = Y ).
+(4)
+By Eq. (3), T2W attention finds the salient parts for the
+given word at each frame, which enforces Eq. (4) to attend
+over the continuous frames instead of concentrating the at-
+tention only on one or some episodic frames as the previous
+P2W attentions. In this way, the whole T2W block exploits
+more temporal contexts to build vision-language associa-
+tions, which facilitates the video reasoning tasks that usu-
+ally require the correct recognition of the temporal patterns.
+Figure. 3 compares P2W and T2W attentions.
+3.3. Fine-Grained Align-Before-Fuse
+Align-before-fuse strategy [24] pulls close vision and
+language embedding spaces calculated by two single-modal
+encoders, which encourages the subsequent cross-modal
+encoder to build more robust semantic associations. Mo-
+tivated by this, we apply this strategy for helping our asym-
+metric cross-modal encoder learn better connections. This
+strategy contrasts the outputs of two single-modal encoders
+by maximizing the similarity of the paired video-text em-
+beddings and minimizing the similarity if they are not
+paired. Suppose sij denotes the similarity score of the i-
+4
+
+th video and the j-th text, then the contrastive objective is:
+Lc = −
+�
+i
+log
+exp(sii/τ)
+�
+j exp(sij/τ) −
+�
+j
+log
+exp(sjj/τ)
+�
+i exp(sij/τ), (5)
+where τ is the temperature. This loss contains two symmet-
+ric parts, where the left term forces the i-th text embedding
+to be close to the i-th video embedding compared with the
+other texts and the right term has a similar effect.
+The seminal research [24] computes sij by the image and
+text global [CLS] tokens, which only capture the coarse-
+level contexts. However, the critical visual patterns (e.g.,
+the salient objects) only occupy small parts of the image,
+which may be neglected by the global [CLS] token since
+it mean-pools the whole image. Such limitation will be fur-
+ther exacerbated in the videos since the videos contain more
+trivial spatial backgrounds, which may be over-exploited by
+the coarse-level contrastive loss. Then the models trained
+by this loss will fail in distinguishing the videos that have
+similar backgrounds while containing diverse events.
+To amend this limitation, we propose the fine-grained
+align-before-fuse strategy which calculates sij by summing
+the token-wise maximum similarities between the patch and
+the word embeddings. In this way, local details will be am-
+plified to pull close the embedding spaces for helping the
+subsequent cross-modal encoder build more robust associa-
+tions. It requires two steps to get fine-grained sij. Firstly,
+for each video patch embedding vn, we find the word em-
+bedding x∗
+n which has the maximum inner product with vn:
+x∗
+n = argmax
+xm∈X
+vT
+n xm,
+(6)
+where X is the word embedding set. By Eq. (6), a most cor-
+responding word will be assigned to the given video patch.
+Secondly, after finding the maximum similarity word for
+each patch, we calculate the token-wise video-to-text simi-
+larity score sv as:
+sv =
+�
+n
+vT
+n x∗
+n.
+(7)
+Similarly, we can calculate the text-to-video one st as:
+st =
+�
+m
+xT
+mv∗
+m,
+(8)
+where v∗
+m is the video patch which has the maximum inner
+product corresponding to the word xm, which is sketched
+in Figure. 4.
+Lf = −
+�
+i
+log
+exp(sv
+ii/τ)
+�
+j exp(sv
+ij/τ)−
+�
+j
+log
+exp(st
+jj/τ)
+�
+i exp(st
+ij/τ), (9)
+3.4. Training Objectives
+To train TW-BERT, as the green blocks shown in Fig-
+ure 2, we totally use four losses which are masked language
+modeling (MLM), video-text matching (VTM), coarse-
+grained video-text contrastive loss and the novel proposed
+fine-grained video-text contrastive loss.
+Masked language Modeling (MLM) [11,23,25,50]. MLM
+aims to predict the masked word tokens given both the video
+and the text contexts. To get it, we first randomly replace
+the input text tokens with the [MASK] token with a proba-
+bility of 15% and then use the [MASK] embedding output
+from the cross-modal encoder to predict the masked word
+by calculating a cross-entropy loss.
+Video-text Matching (VTM) [11,23,50]. VTM calculates
+whether the given video and text are matched or not. To get
+it, for a given video-text pair, we first randomly replace the
+text with the ones from a different video in the same batch.
+Then we concatenate the video and text [CLS] embeddings
+output from the cross-modal encoder and input the concate-
+nated embedding into a binary classifier to judge whether
+the given video-text pair is matched or not.
+Video-text Contrastive (VTC) [2, 13, 14, 23, 49, 50]. As
+detailed in Section 3, VTC contrasts the outputs of two
+single-modal encoders to pull close their embedding space
+to help the subsequent cross-modal encoder build more
+robust vision-language associations. It contains both the
+coarse-grained (Eq. (5)) and fine-grained (Eq. (9)) terms:
+Lvtc = Lc + Lf.
+(10)
+In the implementation, we follow [24] to use the momentum
+queue as a continuously-evolving teacher to provide more
+negative samples.
+4. Experiments
+4.1. Pre-training Dataset
+Following recent work [2, 13, 14, 23, 49], we pre-train
+TW-BERT on Google Conceptual Captions (CC3M) [45]
+containing 3.3M image-text pairs and WebVid-2M [2] con-
+taining 2.5M video-text pairs.
+For CC3M, the image
+is treated as a one-frame video data during pre-training.
+Note that due to the limited storage and computation re-
+sources, we do not use some much larger datasets like
+HowTo100M [35] containing 136M video-text pairs as [25,
+54].
+Also, we do not distil knowledge from CLIP [39],
+which is pre-trained on 400M image-text pairs, as [33].
+4.2. Downstream Tasks
+Text-to-Video Retrieval.
+(i) MSRVTT contains 10K
+YouTube videos with 200K descriptions. We follow [55] to
+use 9K train+val videos for training and report results on the
+1K test split. (ii) DiDeMo [15] contains 10K Flickr videos
+annotated with 40K sentences. (iii) LSMDC consists of
+118,081 video clips sourced from 202 movies, where the
+validation set and the test set contain 7,408 and 1,000
+videos. (iv) ActivityNet Caption contains 20K YouTube
+videos annotated with 100K sentences.
+The training set
+contains 10K videos, and we use val1 set with 4.9K videos
+to report results. For MSRVTT and LSMDC, we perform
+standard text-to-video retrieval. For DiDeMo and Activi-
+5
+
+tyNet Caption, we concatenate all the text captions in the
+same video as a single query and evaluate paragraph-to-
+video retrieval.
+For two tasks, we measures the perfor-
+mances by average recall at K(R@K) and Median Rank on
+zero-shot and fine-tune setups.
+Video Question Answering. (i) MSRVTT-QA [52] is built
+upon videos and captions from MSRVTT [55], which con-
+tains 10K videos with 243K open-ended questions and 1.5K
+answer candidates. (ii) MSVD [5] contains 50K question-
+answer pairs with 2423 answer candidates. We use standard
+train/val/test splits for the two tasks, and report accuracy.
+4.3. Implementation Details
+We initialize our video encoder by ViT-B/16 [42] and
+the text encoder by the first six BERT-Base layers [8]. For
+the cross-modal encoder, the self-attentions in all 3 cross-
+modal attentions (W2P contains 1 and T2W contains 2) are
+initialized by the last 6 BERT-Base layers [8]. The model is
+trained end-to-end during both pre-training and fine-tuning.
+In pre-training, the feature dimension is set to 256 when
+calculating the contrastive losses and the temperature is set
+to 0.05. For the momentum queue, the momentum value
+is 0.995 and the size of the queue is 65,536. The above
+implementation details follow the recent work [23, 24] for
+a fair comparison. We pre-train the model on CC3M and
+WebVid-2M for 10 epochs on 8 NVIDIA A100 GPUs
+where the batch size is 128. We use AdamW [17] optimizer
+with a weight decay of 0.001 and betas (0.9, 0.98). The
+learning rate is first warmed-up to 1e-4 and then decays fol-
+lowing a linear decay schedule.
+During fine-tuning text-to-video retrieval, we sample 8
+frames per video. The model is trained with both VTC and
+VTM losses, and we obtain similarity scores from the out-
+put of the VTM head during inference. For video ques-
+tion answering, we sample 16 frames per video.
+Since
+MSRVTT-QA and MSVD-QA [52] are open-ended VQA,
+in which the answers are in free-form natural language, it is
+common to convert the task to a classification task by pre-
+dicting the answer’s label. We input the concatenation of the
+video and question [CLS] tokens into a two-layer MLP [8]
+for calculating the cross-entropy loss. All the fine-tuning
+experiments are conducted on 8 NVIDIA V100 GPUs.
+4.4. Ablation Studies
+We conduct comprehensive ablation studies to evaluate
+the effectiveness of the proposed trajectory-to-word (T2W)
+attention, concatenation strategy, and fine-grained align-
+before-fuse (FG-ABF) strategy.
+Comparing Methods. Base: We use a symmetric cross-
+modal encoder that contains patch-to-word (P2W) and
+word-to-patch (W2P) attentions. T2W: We replace P2W
+attention in Base by our T2W attention. ConCat: Since our
+cross-modal encoder is asymmetric, we concatenate the text
+and video [CLS] tokens to calculate the VTM loss, which is
+different from the symmetric ones [11,23,49] that only use
+text [CLS] token. TW-BERT: We additionally apply fine-
+grained align-before-fuse to the baseline ConCat and then
+the integral TW-BERT is built.
+Quantitative Results. Table 1 compares the performances
+of diverse baselines. From this table, we can see that T2W
+outperforms Base, e.g., T2W achieves 1.5% R@5 improve-
+ments on MSRVTT for zero-shot evaluation, which sug-
+gests that our T2W attention can exploit more temporal con-
+texts to better solve video-language tasks. Also, ConCat is
+better than T2W on different tasks, e.g., ConCat achieves
+28.0% of R@1 score in DiDeMo zero-shot text-to-video re-
+trieval task while T2W only has 27.4%. This proves that the
+concatenation strategy is effective in this asymmetric cross-
+modal encoder case.
+Lastly, we can see that TW-BERT
+also beats Concat by using the fine-grained align-before-
+fuse strategy, e.g., TW-BERT achieves 1.6% of R@5 im-
+provements on MSRVTT for zero-shot evaluation, which
+validates the power of this strategy.
+Qualitative Results. We visualize the heat maps of the at-
+tention weights of Base and TW-BERT in Figure 5. We see
+that TW-BERT can implicitly form a trajectory for a given
+query word to avoid over-exploiting the trivial spatial con-
+texts as in Base. For example, in (a), TW-BERT tracks the
+hand of the girl in each frame according to the query word
+“practising” while Base attends to the larger while trivial
+regions about the whole body of the girl. Moreover, in (b),
+the trajectory of the ball is tracked by the query “basketball”
+while Base only focuses on the ball region in the first frame.
+4.5. Comparisons with SOTA
+We compare our TW-BERT with previous methods on
+two frequently applied tasks which are video-text retrieval
+(VDTR) and video question answering (VDQA). Table 2
+and 3 report the performances of VDTR on MSRVTT [55],
+DiDeMo [15], LSMDC [43], and ActivityNet Caption [18],
+respectively, where the former three datasets contain both
+zero-shot and fine-tuning setups and the last one only
+has fine-tuning setup.
+Table 4 reports the VDQA on
+MSRVTT [55] and MSVD [5].
+Among the compared
+models, MILES [14], BridgeFormer [13], OA-Trans [49],
+Clipbert [22] and VIOLET [11] are SOTA models pro-
+posed in recently 1-2 years. Note that VIOLET [11] and
+ALPRO [23] distill knowledge from additional large-scale
+BERTs while we do not. Also, we show the number of pre-
+training video-text pairs in these tables for more clear com-
+parisons.
+From these tables, we can find that when the pre-training
+data is in the same scale, TW-BERT achieves the best
+performance compared with all the other models on both
+VDTR and VDQA. For example, on DiDeMo VDTR, TW-
+BERT outperforms BridgeFormer by 4.8% on R@1, or on
+6
+
+(b) “Two people play basketball”
+(a) “A girl practising her arrow tricks”
+(c) “Adult feeds baby with spoon”
+(d) “The dog runs away and picks up the cone”
+Video
+Frames
+Base
+TW-BERT
+Video
+Frames
+Base
+TW-BERT
+Figure 5. Visualizations of the attention maps from cross-modal encoder. Sample (a) and (b) are from MSRVTT [55], (c) and (d) are from
+DiDeMo [15] retrieval dataset. TW-BERT attends to the patches related to given query word by Trajectory-to-Word attention.
+Table 1. Performances of various baselines. R@K and MedR respectively denote recall (%) with K retrieval efforts and median ranking
+for retrieved videos where higher R@K and lower MedR indicate better performance.
+Method
+MSRVTT-ZS
+DiDeMo-ZS
+MSVD-QA
+R@1↑
+R@5↑
+R@10↑
+MedR↓
+R@1↑
+R@5↑
+R@10↑
+MedR↓
+Acc.
+Base
+25.1
+46.4
+57.3
+7.0
+26.6
+52.8
+62.7
+5.0
+47.4
+T2W
+25.8
+47.9
+58.0
+6.0
+27.4
+53.1
+64.0
+5.0
+48.1
+ConCat
+26.1
+48.5
+58.6
+6.0
+28.0
+52.9
+64.2
+4.0
+48.3
+TW-BERT
+26.4
+50.1
+59.6
+5.0
+28.4
+52.9
+64.5
+4.0
+48.5
+MSVD VDQA, TW-BERT outperforms ALPRO by 2.6%.
+Moreover, compared with the models trained on much more
+data, TW-BERT can still achieve the best performances on
+various tasks, e.g., on LSMDC VDTR, TW-BERT outper-
+forms VIOLET by 8.0% on R@10 or on MSRVTT VDQA,
+TW-BERT outperforms MERLOT [59] by 0.5%.
+Note that the videos in LSMDC are longer than
+MSRVTT and DiDeMo, which means that the videos in
+this dataset contain more temporal contexts than the other
+datasets. Then as shown in Table 2, the improvements of
+TW-BERT over other SOTAs are larger than the improve-
+ments on the other datasets. For example, compared with
+BridgeFormer, TW-BERT has an average 3.4% improve-
+ment in the zero-shot setting, while on DiDeMo dataset, the
+corresponding average improvement over MILES is only
+1.6%. Such comparisons further validate the effectiveness
+of TW-BERT in exploiting temporal contexts.
+Among these SOTAs, only when compared with VIO-
+LET, which uses an additional large-scale model DALL-
+E [40] and 32 more times pre-training data than ours (180M
+VS. 5.5M), TW-BERT cannot comprehensively surpass VI-
+OLET on all the tasks. For example, on MSRVTT zero-shot
+VDTR, VIOLET achieves 0.1% higher than TW-BERT or
+on MSRVTT VDQA, VIOLET achieves 0.3% higher than
+7
+
+ACARTable 2. Experiments of text-to-video retrieval on MSRVTT, DiDeMo and LSMDC datasets. “#PT Pairs” lists the number of video-text
+pairs for pre-training. We show results with zero-shot evaluation (top) and fine-tuning evaluation (bottom).
+Method
+#PT Pairs
+MSRVTT
+DiDeMo
+LSMDC
+R@1↑
+R@5↑
+R@10↑
+MedR↓
+R@1↑
+R@5↑
+R@10↑
+MedR↓
+R@1↑
+R@5↑
+R@10↑
+MedR↓
+ActBERT [62]
+120M
+8.6
+23.4
+33.1
+36.0
+-
+-
+-
+-
+-
+-
+-
+-
+MIL-NCE [35]
+120M
+9.9
+24.0
+32.4
+29.6
+-
+-
+-
+-
+-
+-
+-
+-
+TACo [57]
+120M
+9.8
+25.0
+33.4
+29.0
+-
+-
+-
+-
+-
+-
+-
+-
+VideoCLIP [54]
+110M
+10.4
+22.2
+30.0
+-
+16.6
+46.9
+-
+-
+-
+-
+-
+-
+SupportSet [37]
+120M
+12.7
+27.5
+36.2
+24.0
+-
+-
+-
+-
+-
+-
+-
+-
+HERO [25]
+120M
+16.8
+43.4
+57.7
+-
+-
+-
+-
+-
+-
+-
+-
+-
+VIOLET [11]
+186M
+25.9
+49.5
+59.7
+-
+23.5
+49.8
+59.8
+-
+-
+-
+-
+-
+Frozen [2]
+5.5M
+18.7
+39.5
+51.6
+10.0
+21.1
+46.0
+56.2
+7.0
+9.3
+22.0
+30.1
+51.0
+OA-Trans [49]
+5.5M
+23.4
+47.5
+55.6
+8.0
+23.5
+50.4
+59.8
+6.0
+-
+-
+-
+-
+ALPRO [23]
+5.5M
+24.1
+44.7
+55.4
+8.0
+23.8
+47.3
+57.9
+6.0
+-
+-
+-
+-
+BridgeFormer [13]
+5.5M
+26.0
+46.4
+56.4
+7.0
+25.6
+50.6
+61.1
+5.0
+12.2
+25.9
+32.2
+42.0
+MILES [14]
+5.5M
+26.1
+47.2
+56.9
+7.0
+27.2
+50.3
+63.6
+5.0
+11.1
+24.7
+30.6
+50.7
+TW-BERT
+5.5M
+26.4
+50.1
+59.6
+5.0
+28.4
+52.9
+64.5
+4.0
+14.2
+30.4
+36.0
+28.0
+HERO [25]
+120M
+16.8
+43.4
+57.7
+-
+2.1
+11.4
+36.1
+-
+-
+-
+-
+-
+TACo [57]
+120M
+28.4
+57.8
+71.2
+4.0
+-
+-
+-
+-
+-
+-
+-
+-
+SupportSet [37]
+120M
+30.1
+58.5
+69.3
+3.0
+-
+-
+-
+-
+-
+-
+-
+-
+VideoCLIP [54]
+110M
+30.9
+55.4
+66.8
+-
+-
+-
+-
+-
+-
+-
+-
+-
+VIOLET [11]
+186M
+34.5
+63.0
+73.4
+-
+32.6
+62.8
+74.7
+-
+16.1
+36.6
+41.2
+-
+Clipbert [22]
+5.6M
+22.0
+46.8
+59.9
+6.0
+20.4
+48.0
+60.8
+6.0
+-
+-
+-
+-
+Frozen [2]
+5.5M
+31.0
+59.5
+70.5
+3.0
+31.0
+59.8
+72.4
+3.0
+15.0
+30.8
+39.8
+20.0
+ALPRO [23]
+5.5M
+33.9
+60.7
+73.2
+3.0
+35.9
+67.5
+78.8
+3.0
+-
+-
+-
+-
+OA-Trans [49]
+5.5M
+35.8
+63.4
+76.5
+3.0
+34.8
+64.4
+75.1
+3.0
+18.2
+34.3
+43.7
+18.5
+BridgeFormer [13]
+5.5M
+37.6
+64.8
+75.1
+3.0
+37.0
+62.2
+73.9
+3.0
+17.9
+35.4
+44.5
+15.0
+MILES [14]
+5.5M
+37.7
+63.6
+73.8
+3.0
+36.6
+63.9
+74.0
+3.0
+17.8
+35.6
+44.1
+15.5
+TW-BERT
+5.5M
+38.4
+65.1
+76.6
+3.0
+41.8
+71.1
+81.2
+2.0
+21.0
+38.8
+49.2
+11.0
+Table 3. ActivityNet Caption with fine-tuning setting.
+Method
+#PT Pairs
+R@1↑
+R@5↑
+R@10↑
+MedR↓
+Dense [18]
+-
+14.0
+32.0
+-
+34.0
+FSE [60]
+-
+18.2
+44.8
+-
+7.0
+CE [28]
+-
+18.2
+47.7
+-
+6.0
+HSE [60]
+-
+20.5
+49.3
+-
+-
+Clipbert [22]
+5.6M
+21.3
+49.0
+63.5
+6.0
+TW-BERT
+5.5M
+31.7
+62.3
+74.9
+3.0
+Table 4. Experiments of video question answering on MSRVTT
+and MSVD datasets in top-1 accuracy (%).
+Method
+#PT Pairs
+MSRVTT
+MSVD
+Clipbert [22]
+5.6M
+37.4
+-
+ALPRO [23]
+5.5M
+42.1
+45.9
+SINGULARITY [21]
+5.5M
+42.7
+45.9
+LGDN [30]
+15.2M
+43.1
+-
+SSML [1]
+100M
+35.1
+35.1
+JustAsk [56]
+69M
+41.5
+46.3
+MERLOT [59]
+180M
+43.1
+-
+VIOLET [11]
+186M
+43.9
+47.9
+TW-BERT
+5.5M
+43.6
+48.5
+TW-BERT, while such marginal improvements are got at
+the cost of much more training resources.
+Furthermore,
+TW-BERT still outperforms VIOLET on the other tasks,
+e.g., on DiDeMo VDTR, TW-BERT outperforms VIOLET
+by 9.2% on R@1 or on MSVD VDQA, TW-BERT outper-
+forms VIOLET by 0.6%. These comparisons confirm the
+effectiveness of the proposed TW-BERT.
+5. Conclusion and Limitation
+We propose a novel Trajectory-Word BERT (TW-
+BERT) that builds Trajectory-to-Word alignments for solv-
+ing video-language tasks. In particular, we introduce an
+asymmetric cross-modal encoder which contains word-to-
+patch (W2P) and Trajectory-to-Word (T2W) to capture
+cross-modal associations. Moreover, the fine-grained con-
+trastive loss is proposed to amplify the local details for help-
+ing the subsequent cross-modal encoder build more robust
+connections.
+Extensive experiments across diverse tasks
+confirm the effectiveness of the proposed TW-BERT.
+There are two limitations which may limit the model’s
+effectiveness.
+Firstly, as Table 2 shows, various VDL-
+BERTs train their model by 120M video-text pairs while
+we only use 5.5M. Secondly, we do not consider the tra-
+jectory characteristic in the video encoder while only in
+the cross-modal, which may miss certain temporal contexts.
+We do this since the words contain integral semantic knowl-
+edge that can help efficiently locate the trajectory, while if
+we consider the trajectory characteristic in the video en-
+coder, each pixel/patch should be tracked across the whole
+video, which has higher computation complexity. To solve
+two limitations, the ideas of efficient Transformers [36,38]
+can be brought in our TW-BERT to reduce the computation
+8
+
+complexity and we will also extend our computation power
+like the GPU servers to train a trajectory-based video en-
+coder.
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+Intrinsic interface adsorption drives selectivity in atomically smooth nanofluidic channels
+Phillip Helms,1, 2 Anthony R. Poggioli,1, 3 and David T. Limmer1, 2, 4, 3, ∗
+1Department of Chemistry, University of California, Berkeley, California 94720, USA
+2Chemical Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
+3Kavli Energy NanoScience Institute, Berkeley, California 94720, USA
+4Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
+(Dated: January 12, 2023)
+Specific molecular interactions underlie unexpected and useful phenomena in nanofluidic systems, but re-
+quire descriptions that go beyond traditional macroscopic hydrodynamics. In this letter, we demonstrate how
+equilibrium molecular dynamics simulations and linear response theory can be synthesized with hydrodynam-
+ics to provide a comprehensive characterization of nanofluidic transport. Specifically, we study the pressure
+driven flows of ionic solutions in nanochannels comprised of two-dimensional crystalline substrates made from
+graphite and hexagonal boron nitride. While simple hydrodynamic descriptions do not predict a streaming elec-
+trical current or salt selectivity in such simple systems, we observe that both arise due to the intrinsic molecular
+interactions that act to selectively adsorb ions to the interface in the absence of a net surface charge. Notably,
+this emergent selectivity indicates that these nanochannels can serve as desalination membranes.
+Recent advances in nanoscale fabrication techniques have
+enabled the synthesis of nanofluidic systems with novel
+functionalities, [1–3] with applications to biotechnology [4],
+filtration [5–7], and computation [8–10].
+For example,
+nanofluidics-based membranes have leveraged atomic level
+details like those of evolved biological membranes [11–18]
+to circumvent traditional trade-offs between permeability and
+selectivity that plague membrane technology [19–22]. While
+continuum-level hydrodynamic descriptions can remain accu-
+rate at scales of a few nanonmeters, enabling some general de-
+sign principles to be deduced [23–26], the continued develop-
+ment of nanofluidic devices is limited by a lack of understand-
+ing of emergent interfacial effects which are resolutely molec-
+ular in origin. With large surface to volume ratios, the prop-
+erties of fluids confined to nanometer scales are determined in
+large part by a delicate interplay of interactions between the
+bounding surfaces and the working fluid. To understand and
+design nanofluidic devices, an approach that combines macro-
+scopic and molecular perspectives is necessary [27].
+In this letter, we show how interfacial atomic structure
+affects the directed transport of an electrolyte solution in
+nanochannels made of atomically flat graphite (GR) and
+hexagonal boron nitride (BN) walls using molecular dynamics
+simulations unified with a contemporary perspective on hydro-
+dynamics. These simple nanofluidic systems have been stud-
+ied extensively because of their intriguing transport properties,
+such as anomalously high permeabilities in GR [28–36], and
+the potential to augment their functionality with selectivity for
+desalination or blue energy applications [37–46]. By com-
+puting the spatially-resolved volumetric, charge, and species
+transport coefficients from equilibrium correlations [47–49]
+we elucidate the importance of specific molecular interactions
+on nanofluidic device functionality.
+While from a contin-
+uum perspective, driving the solution with a pressure gradient
+should result in salt filtration or electric current only when the
+confining walls have a net charge, we discover that the intrinsic
+interfacial adsorption of ions can lead to streaming electrical
+currents and a novel, emergent desalination mechanism.
+We focus on the two systems illustrated in Fig. 1(a), consist-
+ing of an aqueous solution of potassium chloride confined in
+nanochannels with fixed walls of either BN or GR. Because of
+the experimental similarity between the structure of BN and
+GR lattices, we spaced atoms and lattice layers identically,
+with interatomic and interlayer spacings of 1.42 Å and 3.38
+Å [50, 51]. Each wall has three layers, using AA’ and AB
+stacking for BN and GR, respectively, to match their equilib-
+rium structures, with lattice unit cells repeated 8 and 13 times
+in the 푥 and 푦 directions for a cross-sectional surface area of
+nearly 9 nm2. The walls were separated such that the spacing
+between the center of mass of the innermost wall layers was
+퐻 ≈ 5.7 nm, with the channel width adjusted to ensure a bulk
+water density of ̄휌w ≈ 1 g∕cm3. The channels were filled with
+푁w = 1920 TIP4P/2005 water molecules with rigid geome-
+tries imposed using the SHAKE algorithm [52, 53], 푁K+ = 40
+potassium ions and 푁Cl− = 40 chloride ions, resulting in a
+nearly 1 M electrolyte solution.
+We
+evolved
+this
+system
+according
+to
+underdamped
+Langevin dynamics,
+푚푖 ̇퐯푖 = −휁푖퐯푖 + 퐅푖
+(퐫푁) + 퐑푖
+(1)
+where each particle 푖 has mass 푚푖, velocity 퐯푖, and experi-
+ences a friction 휁푖, with forcing from interparticle interac-
+tions 퐅푖
+(퐫푁), and random noise 퐑푖. The random force is a
+Gaussian random variable with mean ⟨푅푖,훼⟩ = 0 and variance
+⟨푅푖,훼(푡)푅푖′,훼′(푡′)⟩ = 2푘B푇 휁푖훿푖,푖′훿훼,훼′훿(푡−푡′) for each cartesian
+coordinate 훼, where 푘B푇 is Boltzmann’s constant times tem-
+perature. Periodic boundary conditions were imposed in all
+three spatial dimensions, with a vacuum layer in the 푧 direction
+of 5 nm to ensure no interaction between periodic images of
+the channel. Intermolecular Lennard-Jones forces were chosen
+from literature-reported values to reproduce the solubility of
+ions in water and match the ab initio equilibrium fluid structure
+in BN and GR nanochannels [54, 55], with Lorentz-Berthelot
+mixing rules defining heteroatomic interactions.
+Addition-
+ally, water molecules, charged ions, and the BN wall atoms
+interacted with Coulomb potentials, where boron and nitro-
+arXiv:2301.04231v1  [cond-mat.mes-hall]  10 Jan 2023
+
+2
+FIG. 1. Description of the systems considered and resulting equilib-
+rium density distributions. (a) A snapshot of the nanochannels con-
+sidered with the left (right) side corresponding to the boron nitride
+(graphite) nanochannel. The top images show the wall structure, with
+each wall composed of three layers and the periodic unit cell outlined
+in red. (b) The molecular species density distributions for potassium
+(green), chloride (purple), and water (black) as a function of position,
+normalized by bulk densities.
+gen atoms have charges of ± 1.05e, with e being the elemen-
+tary charge, using an Ewald summation as implemented in
+LAMMPS [56]. For all data presented here, we performed
+5 independent simulations, each starting with an equilibration
+run for 5 ns with 푚푖∕휉푖 = 2 ps, followed by a production run for
+10−20 ns with 푚푖∕휉푖 = 10 ns at a temperature of 298 K. In all
+plots, lines represent averages and error bars represent the stan-
+dard deviation for the 5 simulations. All scripts used to pro-
+duce these results and the raw data are openly available [57].
+Figure 1(b) shows the equilibrium particle number densi-
+ties, 휌푖(푧), for all species, 푖 = {w, K+, Cl−}, in the BN and
+GR channels, relative to their bulk values, ̄휌푖. We observe
+similar structures in both materials with interfacial layering
+of water that is consistent with previous simulations of neat
+water[35, 55]. The distribution of ions near such interfaces
+is known to be highly dependent on ion species, and the pro-
+files shown are consistent with previous simulations [58–60]
+A dense layer of pure water accumulates near the wall, with the
+molecules oriented such that they induce a small local negative
+charge. The next layers are enriched in alternating concentra-
+tions of potassium and chloride ions, with depletion (accumu-
+lation) of water molecules accompanying potassium (chloride)
+enrichment. The two materials differ slightly, with a higher
+water density in the first layer of BN resulting in layering with
+higher amplitude in BN compared to GR, though in both sys-
+tems the layering in the density decays to its bulk value for
+each species, ̄휌푖, within 1.5 nm.
+We consider fluxes induced by a pressure differential,
+−Δ푃푥, imposed electrostatic potential drop −ΔΦ푥, or water
+chemical potential differential, −Δ휇푥, with subscripts denot-
+ing application in the 푥 direction parallel to the walls, and limit
+ourselves to small driving strengths. In this limit, linear re-
+sponse theory dictates that induced local fluxes are linearly
+dependent on driving forces,
+⎛
+⎜
+⎜⎝
+푞(푧)
+푗(푧)
+푑(푧)
+⎞
+⎟
+⎟⎠
+=
+⎛
+⎜
+⎜⎝
+푞푄 푞퐽
+푞퐷
+푗푄 푗퐽
+푗퐷
+푑푄 푑퐽 푑퐷
+⎞
+⎟
+⎟⎠
+⎛
+⎜
+⎜⎝
+−Δ푃푥
+−ΔΦ푥
+−Δ휇푥
+⎞
+⎟
+⎟⎠
+,
+(2)
+where 푞(푧) is the volumetric flow, 푗(푧) the charge flux, 푑(푧)
+the excess water flux, and the 푎퐵(푧) are the spatially de-
+pendent mobilities. The excess water flux 푑(푧) represents the
+local water flux relative to what would be predicted from the
+bulk water density and the local total flux of water and ions,
+and it is considered here because it is particularly relevant for
+desalination. The diagonal elements of the mobility matrix
+link a given forcing directly to its conjugate flux – e.g., 푗퐽
+links the potential drop −ΔΦ푥 directly to the induced charge
+flux 푗(푧) – while the off-diagonal elements are the so-called
+cross-terms linking, for example, an induced charge flux to an
+applied pressure differential. The total fluxes include the to-
+tal volumetric flow 푄, charge flux 퐽, and excess water flux 퐷.
+We index mobilities by the local induced flux 푎 and total flux
+퐵 directly conjugate to a particular forcing.
+The local fluxes are defined microscopically as
+푞(푧, 푡) = 퐻
+푁
+푁
+∑
+푖=1
+푣푖,푥(푡)훿 [푧 − 푧푖(푡)]
+푗(푧, 푡) = 1
+퐴s
+푁
+∑
+푖=1
+푐푖푣푖,푥(푡)훿 [푧 − 푧푖(푡)]
+푑(푧, 푡) = 1
+퐴s
+푁
+∑
+푖=1
+푣푖,푥(푡) (훿푖,w − 푓 b
+w
+) 훿 [푧 − 푧푖(푡)]
+(3)
+where particle 푖 has velocity 푣푖,푥(푡) and position 푧푖(푡) at time
+푡, a static charge of 푐푖, and 훿푖,w is a Kroniker delta that re-
+turns 1 if particle 푖 is a water molecule and is 0 otherwise.
+The bulk mole water fraction is defined as 푓 b
+w = 푁b
+w∕푁b,
+where 푁b
+w and 푁b are respectively the average numbers of wa-
+ter molecules and all molecules in the bulk and 퐴s is the sur-
+face area associated with the fluid-wall interface. The spatial
+dependence can be integrated out by defining total fluxes, such
+as 푄 = 1∕퐻 ∫ 퐻
+0
+푑푧 푞(푧), with analogous definitions for 퐽 and
+퐷. Total channel conductivities can be evaluated as 퐴퐵 =
+1∕퐻 ∫ 퐻
+0
+푑푧 푎퐵(푧), resulting in total flux linear response re-
+lations such as 푄 = −푄푄Δ푃푥 −푄퐽ΔΨ푥 −푄퐷Δ휇푥. While
+
+a
+Boron Nitride (BN
+Graphite (GR
+b)
+2
+2
+3
+4
+5
+nm
+; (
+nm3
+the integrated conductivities must obey Onsager reciprocal re-
+lations, 퐴퐵 = 퐵퐴, mobilities are under no such constraint.
+It is possible for 푎퐵(푧) ≠ 푏퐴(푧).
+Rather than attempting to calculate mobilities directly via
+nonequilibrium simulations, we use fluctuation-dissipation re-
+lations in order to obtain transport coefficients from equilib-
+rium flux correlations [47–49]. This allows us to avoid run-
+ning separate nonequilibrium simulations for each term in the
+mobility matrix, and ensures the validity of linear response.
+We adopt the Einstein-Helfand approach over the Green-Kubo
+method, as recent work has demonstrated its enhanced sta-
+tistical efficiency [47]. Mobilities are obtained as the long
+time slope of the correlation between time-integrated local and
+global fluxes
+푎퐵 =
+푉
+2푘B푇 lim
+푡→∞
+휅푎퐵(푡)
+푡
+,
+(4)
+with the correlation function
+휅푎퐵 = ∫
+푡
+0
+푑푡′
+∫
+푡
+0
+푑푡′′ ⟨푎(푧, 푡′) 퐵(푡′′)⟩ ,
+(5)
+volume
+푉
+=
+퐴s퐻,
+and
+brackets
+representing
+an
+equilibrium
+average.
+Similarly,
+conductivities
+can
+be
+obtained
+using
+correlations
+between
+global
+fluxes,
+퐴퐵
+=
+(푉 ∕2푘B푇 ) lim푡→∞ 퐾퐴퐵(푡)∕푡
+with
+퐾퐴퐵 = ∫ 푡
+0 푑푡′ ∫ 푡
+0 푑푡′′ ⟨퐴(푡′)퐵(푡′′)⟩.
+Previous work has demonstrated that while equilibrium
+structures suggest only minor differences between water in BN
+and GR nanochannels, the dynamics of the confined fluid are
+strikingly different. This results in large differences in the fric-
+tion between the fluid and walls, and significant differences
+in resultant channel permeabilities.
+[28, 35, 61, 62]. In the
+presence of ions, the interfacial structure of water is altered
+and as a consequence the friction may change. In Fig. 2 (a),
+we show the integrated global flux correlation function 퐾푄푄
+as a function of time for both nanochannels. After approx-
+imately 200 ps, the correlation functions approach a linear
+dependence on time and their slopes give the hydraulic con-
+ductivities as [BN]
+푄푄
+= 18.0 ± 9.2 mol nm5 kJ−1 ns−1 and
+[GR]
+푄푄 = 106 ± 40 mol nm5 kJ−1 ns−1, which differ by nearly
+an order of magnitude.
+While the hydraulic conductivities deduced above are inde-
+pendent of a specific hydrodynamic model, they can be con-
+nected to continuum theory through the slip length 푙s. In con-
+trast to the no-slip condition typically applied in macroscopic
+contexts, which specifies that the fluid velocity exactly van-
+ishes at the walls, the small confinement scales and enhanced
+importance of interfacial details in nanofluidic applications
+typically require application of the finite-slip condition. This
+condition specifies that the velocity at the wall is proportional
+to the shear strain at the wall, 푣푥 = 푙s(휕푣푥∕휕푧)|푧=0. The slip
+length is interpreted geometrically as the distance beyond the
+interface where the extrapolated flow profile is zero, as illus-
+trated in Fig. 2 (b).
+FIG. 2. Comparison of the hydraulic conductivity and slip length for
+the GR (black) and BN (blue) nanochannels. (a) The time-integrated
+global flux correlation function 퐾푄푄 versus time. (b) Comparison
+of the slip lengths for both materials, computed from the hydraulic
+conductivity (dark), against previously reported results for neat water
+(light) [28]. The inset illustrates the geometric interpretation of the
+slip length.
+To apply a hydrodynamic interpretation, we consider only
+the region where a hydrodynamic description is expected to be
+valid by defining the effective hydrodynamic interface as the
+the location of the second water density peak in Fig. 1(b) [26].
+At this distance, microscopic density correlations have de-
+cayed and the fluid is well described as a continuous medium.
+The Poiseuille solution for the hydraulic mobility in the pres-
+ence of a finite slip length is given by
+푞푄(푧) =
+퐻2
+hyd
+2휂
+[
+푙s
+퐻hyd
++
+푧
+퐻hyd
+−
+푧2
+퐻2
+hyd
+]
+,
+(6)
+where 퐻hyd is the distance between hydrodynamic interfaces,
+and 휂 is the estimated viscosity of the solution. This expression
+may be integrated to determine the hydraulic conductivity
+푄푄 =
+퐻2
+hyd
+12휂
+(
+1 + 6
+푙s
+퐻hyd
+)
+,
+(7)
+which allows us to relate the measured values of 푄푄 in GR
+and BN to the corresponding slip lengths provided 휂 is known.
+Here, we use a viscosity of 휂 = 1.0 mPa s, obtained by interpo-
+lating literature values for this electrolyte model [54]. Figure 2
+
+a)
+6
+2
+nm
+2
+0
+0
+100
+200
+300
+400
+t/ps
+b)
+40
+(2) 7
+nm
+20
+0
+BN
+GR4
+FIG. 3. Pressure driven hydraulic (a), streaming (b), and excess wa-
+ter (c) mobility profiles for BN (left, blue) and GR (right, black).
+The red shaded regions demarcate areas where hydrodynamics are
+invalidated. In (a), the red dashed curve corresponds to the hydrody-
+namic estimate from the hydraulic conductivity. In (b) and (c), the
+red dashed curves are the mobility predictions from the product of
+the hydraulic mobility and appropriate density.
+(b) indicates the resulting slip lengths, 푙[BN]
+s
+= 4.0 ± 2.5 nm
+and 푙[GR]
+s
+= 27 ± 10 nm, and compares them against previ-
+ously reported results for neat water [28]. With the slip in
+GR nanochannels being approximately an order of magnitude
+larger than the slip in BN nanochannels, it is clear that the
+qualitative results do not change significantly with the addi-
+tion of salt. The material-dependency of 푙s has been observed
+in various contexts experimentally [32, 63–66] and is gen-
+erally understood to arise from a decoupling of structure and
+dynamics, though the precise physical mechanism is debated
+[28, 35, 61, 67–69]. Quantitatively, our simulations also sug-
+gest a decrease in slip as salt is added, which is consistent with
+other observations for slip on hydrophobic surfaces, where
+increasing fluid-wall friction results as a consequence of en-
+hanced equilibrium force fluctuations from the heterogeneous
+solution. [70–72].
+More detailed insight into the differences in transport char-
+acteristics between BN and GR nanochannels can be obtained
+by computing the spatially-dependent hydraulic mobility us-
+ing Eq. 4. The results of this calculation for GR and BN are
+shown in Fig 3 (a). We also show the hydrodynamic mobility
+profiles calculated from Eq. 6 for comparison to the macro-
+scopic theory. As expected for the conductivity, we observe
+approximately an order of magnitude difference between the
+peaks in the hydraulic mobilities in the BN and GR nanochan-
+nels. The mobility profile is nearly flat for GR and exhibits a
+slight curvature for BN, indicative of the differences in slip. In
+the boundary region, the mobility profile qualitatively mimics
+the fluid density profile with greater (lesser) flux coinciding
+with density peaks (troughs).
+We find that the molecular interfacial structure also affects
+the cross-terms in the mobility matrix in Eq. 2. The stream-
+ing mobility 푗푄, which quantifies the electrical current pro-
+file produced by applying a pressure differential, is shown in
+Fig 3(b) for both systems. We observe the emergence of three
+layers of electrical current of alternating sign near the fluid
+wall boundary, and no net current in the bulk of the chan-
+nel. Because the applied pressure produces particle flux in all
+nanochannel regions, the alternating current is caused by ion
+density localization at the interface, with positive (negative)
+current where potassium (chloride) ions are enriched. These
+interfacial effects decay away from the wall more slowly than
+those observed with the hydraulic mobility, with net charge
+flux penetrating into the hydrodynamic region defined by the
+hydraulic mobility.
+By integrating the mobility across the
+channel, we find that the streaming conductivity 퐽푄 is statis-
+tically indistinguishable from zero for both materials, indicat-
+ing no net ionic transport. Though not shown, our calculations
+verify the lack of symmetry between cross-term mobilities,
+with 푞퐽 being zero at all points in the channel, within sta-
+tistical accuracy, consistent withq 푞퐽 ≠ 푗푄 while main-
+taining 푄퐽 = 퐽푄.
+The pressure driven excess water mobility 푑푄, is shown
+in Fig. 3(c) as computed using Eq. 4 for both materials. This
+quantity is directly related to the desalination capabilities of
+a nanofluidic channel, and its magnitude determined by the
+channel’s selectivity and permeability. This transport is sum-
+marized by the integrated mobility, 푑푄, with 푑푄 > 0 cor-
+responding to the selective flux of water through the channel.
+We find a positive integrated value 푑푄 > 0 for both materials,
+demonstrating a preferential water selectivity and correspond-
+ing salt rejection capability.
+The spatial dependence of the cross-term mobility pro-
+files can be understood via a combination of microscopic and
+macroscopic perspectives.
+The streaming mobility may be
+evaluated microscopically as a product of the local density pro-
+files and the hydraulic mobility. For the streaming mobility
+this is, 푗푄(푧) = [휌K+(푧) − 휌Cl+(푧)]∕휌tot(푧)푞푄(푧)푁∕푉 ,
+where 휌tot(푧) = 휌w(푧) + 휌K+(푧) + 휌Cl−(푧). Though a common
+decomposition in macroscopic hydrodynamics, this is a non-
+trivial statement when considering the microscopic mobilities.
+The red dashed line in Fig. 3(b) shows this estimate agrees
+
+a)
+nm
+ps
+0.2
+mol
+kJ
+0.1 :
+ob1
+M
+0.0
+2
+0
+1
+3
+5
+4
+b)
+0.1
+e
+ps
+0.0
+M
+-0.1
+０
+2
+3
+1
+4
+5
+c)
+10
+ps
+5
+1dQ
+M
+0
+0
+2
+3
+1
+4
+5
+nm
+nm5
+FIG. 4. Estimates of (a) hydraulic conductivity and (b) water selec-
+tivity in simple GR (black) and BN (blue) nanochannels versus chan-
+nel height 퐻. Red shaded regions indicate channel heights where
+boundary effects from confining walls interact, meaning our estimate
+is most reliable for 퐻 ≳ 2 nm.
+well with estimate using Eq. 4. The same functional decom-
+position holds for the excess water flux, which can be obtained
+from the product of the hydraulic mobility and the excess water
+density 푑푄(푧) = (휌w(푧)∕휌tot(푧) − ̄휌w∕ ̄휌tot
+) 푞푄(푧)푁∕푉 .
+This decomposition is shown in the red dashed line in
+Fig. 3(c). Both of these decompositions follow directly from
+the Langevin equations of motion. While the excess water mo-
+bilities for both materials are qualitatively similar because of
+qualitatively similar equilibrium density distributions and hy-
+draulic mobility profiles, the quantitative difference arises due
+to the differences in magnitude of the hydraulic conductivity.
+The first contact layer is nearly salt free, so while interfacial
+friction slows pressure driven transport, the high water purity
+gives a large peak in excess water mobility. There is a sec-
+ond excess water mobility peak near the second water density
+peak. The enrichment and depletion of chloride and potas-
+sium, respectively, brings the overall salt density close to its
+bulk value and leaves an excess concentration of water where
+the hydraulic mobility also peaks.
+The molecular dynamics calculations suggest that the trans-
+port properties of the nanochannel can be decomposed as a
+sum of a molecular interfacial component, and a continuum
+bulk component. The interfacial component depends sensi-
+tively on specific molecular interactions as they manifest in
+non-uniform density profiles. Beyond the domain of those
+density correlations, which for these channels extend around
+2 nm into the channel, the transport is well described by
+Poiseuille flow with a large slip length. This decomposition
+allows us to infer the height dependence of the channel’s se-
+lectivity and permeability. We can calculate the size depen-
+dent conductivity using an integrated mobility 푄푄(퐻) =
+2 ∫ 퐻∕2
+0
+푑푧 푞푄(푧)∕퐻, where we employ the inversion sym-
+metry of the channel to integrate over only half of the channel.
+These conductivities are shown for BN and GR in Fig. 4(a)
+normalized against [GR]
+푄푄 . The red regions in Fig. 4 indicate
+system sizes which would lead to overlapping interfacial re-
+gions, for which our decomposition is not anticipated to be
+valid.
+Because the hydraulic mobility profile is nearly flat
+in the hydrodynamic region, which is expected when 푙푠 ≫
+퐻hyd∕6, the overall permeability increases linearly with chan-
+nel height, which is not as fast as anticipated from traditional
+hydrodynamics with a no-slip boundary condition.
+A similar approach can be used to compute the depen-
+dency of the water selectivity on the height of the chan-
+nel.
+To compute the selectivity, we first can determine
+a pressure driven salt mobility s푄(푧)
+=
+[휌K+(푧) +
+휌Cl−(푧)]∕휌tot(푧)푞푄(푧)푁∕푉 . The ratio of salt to total par-
+ticle flux as a function of channel height is obtained as
+푓salt(퐻) =
+∫ 퐻∕2
+0
+푑푧 s푄(푧)
+푁
+푉 ∫ 퐻∕2
+0
+푑푧 푞푄(푧)
+(8)
+which is shown in Fig. 4(b) normalized against the overall
+number fraction of ions in the bulk, ̄푓salt = ( ̄휌K+ + ̄휌Cl−)∕ ̄휌tot.
+This provides a direct measurement of the size dependence
+of the nanochannel selectivity. Consistent with the inference
+from the excess water mobility, the salt flux is supressed rela-
+tive to its expected value from the bulk concentration of ions
+and the total channel conductivity. We find that BN and GR
+nanochannels have effectively identical selectivities, primar-
+ily because of their similar equilibrium fluid density distribu-
+tions and qualitatively similar hydraulic mobility profiles. For
+the nanochannel size and ion concentrations considered here,
+the flux of salt ions is reduced by approximately 25%, while
+shrinking the nanochannel until interfacial regions overlap at
+around 2 nm could provide a reduction of around 50%. Due to
+the intrinsic interfacial absorption of ions to the interface and
+their resultant suppressed mobility, as the nanochannel size is
+decreased its selectivity is enhanced. An optimal desalination
+device must separate ions from water with both high selectiv-
+ity as well as high permeability, and these phenomenological
+channel scaling observations suggests that for both BN and GR
+this optimum is between 2 and 5 nm.
+This mechanism of selective transport, and the ability of
+the channel to separate salt from water, is a result of an in-
+terplay between local molecular interactions that drive ions to
+the fluid-solid boundary in the absence of a net surface charge
+of the substrate. These molecular interfacial features estab-
+lished a nonuniform fluid composition across the channel that,
+when combined with a spatially resolved evaluation of the hy-
+draulic mobilities, provide a complete description of the trans-
+port within the nanochannel. The promise of this mechanism
+for desalination technology is strikingly enhanced when this
+water selectivity is coupled with the anomalously high perme-
+ability of GR nanochannels. This framework is general and
+
+a)
+1.0
+8
+L
+0.5
+可Q
+L
+0.0
+b)
+1.0
+Tsalt
+0.5
+Jsalt
+4
+0.0
+0
+1
+2
+3
+4
+5
+H 
+nm6
+can be used to understand and engineer other functionality in
+nanofluidic systems. Employing recent generalizations of re-
+sponse theory,[73–75] our approach could be extended outside
+the regime of linear response to provide insight into perfor-
+mance at high driving strengths and between multiple driving
+forces.
+Acknowledgments – This study is based on the work sup-
+ported by the U.S. Department of Energy, Office of Science,
+Office of Advanced Scientific Computing Research, Scientific
+Discovery through Advanced Computing (SciDAC) program,
+under Award No. DE-AC02-05CH11231. A. R. P was also
+supported by the Heising-Simons Fellowship from the Kavli
+Energy Nanoscience Institute at UC Berkeley and D. T. L ac-
+knowledges support from the Alfred P. Sloan Foundation.
+Data availability – The source code for the calculations
+done and all data presented in this work are openly available
+on Zenodo at https://doi.org/10.5281/zenodo.7522996 [57]
+∗ dlimmer@berkeley.edu
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+

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+Nonreciprocal Cahn-Hilliard equations emerging as one of eight universal amplitude equations
+Tobias Frohoff-H¨ulsmann1, ∗ and Uwe Thiele1, 2, †
+1Institut f¨ur Theoretische Physik, Westf¨alische Wilhelms-Universit¨at M¨unster, Wilhelm-Klemm-Str. 9, 48149 M¨unster, Germany
+2Center for Nonlinear Science (CeNoS), Westf¨alische Wilhelms-Universit¨at M¨unster, Corrensstr. 2, 48149 M¨unster, Germany
+Oscillatory behavior is ubiquitous in out-of-equilibrium systems showing spatio-temporal pattern formation.
+Starting from a large-scale linear oscillatory instability – a conserved-Hopf instability – that naturally occurs in
+systems with two conservation laws, we derive the corresponding amplitude equation. This completes the set
+of such universal equations for the eight types of instabilities in homogeneous isotropic systems resulting from
+the combination of three features: large-scale vs. small-scale instability, stationary vs. oscillatory instability, and
+instability without and with conservation law(s). The derived universal equation generalizes a phenomenological
+model of considerable recent interest, namely, the nonreciprocal Cahn-Hilliard equation, and plays a similar role
+in the systematics of pattern formation as the complex Ginzburg-Landau equation.
+The concept of active matter emerged as a paradigm in the
+description of a wide variety of biochemophysical phenomena
+on multiple scales ranging from the collective behaviour of
+molecules within biological cells to the dynamics of tissues or
+human crowds [14]. In a narrow interpretation, active matter
+always involves chemo-mechanical coupling and shows some
+kind of self-sustained (collective) motion of the microscopic
+ingredients [7, 33, 48, 51]. In a wider sense, active systems
+encompass open systems that are kept out of equilibrium by
+a troughflow of energy [35], and therefore may develop self-
+organized patterns of states and/or motion. This then includes
+the large spectrum of systems described by reaction-diffusion
+models [28, 30, 44] and systems characterized by the interplay
+of phase separation and chemical reactions [5].
+In this context, predator-prey-type nonreciprocal interac-
+tions between constituents of active matter have recently be-
+come a particular focus as the implied breaking of Newton’s
+third law results in a rich spectrum of nascent self-excited dy-
+namic behaviour [10, 17, 23, 29, 36]. Beside various (stochas-
+tic) agent-based models of Langevin-type also continuous de-
+terministic field theories have been proposed, most notably, in
+the form of nonreciprocal Cahn-Hilliard models [16, 47, 56].
+The latter add nonreciprocal interactions to classical Cahn-
+Hilliard models [9] (model-B in [20]) that describe the dy-
+namics of phase-separation e.g. in binary or ternary mixtures
+[32, 37].
+In particular, the resulting nonreciprocal Cahn-
+Hilliard models represent two conservation laws with nonva-
+riational coupling. It is shown that this coupling may result in
+coarsening traveling and oscillatory states [16, 47, 56], arrest
+or suppression of coarsening [16], formation of small-scale
+spatial (Turing) patterns as well as stationary, traveling and
+oscillatory localised states [15] - all features forbidden in stan-
+dard reciprocal Cahn-Hilliard models.
+However, the nonreciprocal Cahn-Hilliard model is intro-
+duced on phenomenological grounds by symmetry considera-
+tions, but no derivation of the field theory from a microscopic
+description or other deeper justification has been provided yet.
+Here, we show that the model indeed merits extensive study
+∗ t froh01@uni-muenster.de; ORCID ID: 0000-0002-5589-9397
+† u.thiele@uni-muenster.de; http://www.uwethiele.de; ORCID ID: 0000-
+0001-7989-9271
+as it actually represents one of the universal equations of pat-
+tern formation. One may even argue that it corresponds to
+the “last missing amplitude equation” of the basic eight types
+of linear instabilities in spatially extended systems. An am-
+plitude (or envelope) equation describes the universal spatio-
+temporal dynamics of the essential linear mode(s) in the vicin-
+ity of a stability threshold, and can be systematically derived
+in a weakly nonlinear approach [21]. The mentioned eight in-
+stability types result from the combination of three features:
+(i) large-scale vs. small-scale instability, (ii) stationary vs. os-
+cillatory instability, and (iii) instability without and with con-
+servation law(s). The spatial and temporal character of an in-
+stability encoded in features (i) and (ii) is well captured in the
+classification of instabilities by Cross and Hohenberg [11],
+and the four corresponding amplitude equations for systems
+without conservation law are very well studied. One exam-
+ple is the complex Ginzburg-Landau equation [1] valid near
+the onset of a large-scale oscillatory (aka Hopf or type IIIo
+[11]) instability. An overview of the basic eight instability
+types, their dispersion relations and the seven existing ampli-
+tude equations is provided in section 1 of the Supplementary
+Material.
+However, the consequences of conservation laws in the
+full range of pattern-forming systems are less well studied:
+Small-scale stationary and oscillatory cases with a conserva-
+tion law are considered in [34] and [53], respectively, with
+applications to pattern formation in the actin cortex of motile
+cells [6, 55], in crystallization [50], and in magnetoconvec-
+tion [27]. However, only recently it was shown that the stan-
+dard single-species Cahn-Hilliard equation does not only de-
+scribe phase separation in a binary mixture but furthermore
+represents the amplitude equation valid in the vicinity of a
+large-scale stationary instability in a system with a conser-
+vation law [2]. In consequence, close to onset, a reaction-
+diffusion system with one conservation law, as e.g. discussed
+in [4, 6, 8, 13, 19, 22, 52], can be quantitatively mapped onto
+a Cahn-Hilliard equation. Similarly, it captures core features
+of certain collective behavior in chemotactic systems [46] and
+for cell polarization in eukaryotic cells [3].
+This leaves only one of the eight cases unaccounted for,
+namely, the large-scale oscillatory instability with conserva-
+tion laws, that we call here conserved-Hopf instability. In
+the following, we consider active systems with two conser-
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+arXiv:2301.05568v1  [nlin.PS]  13 Jan 2023
+
+2
+vation laws and show that the corresponding universal ampli-
+tude equation is a general nonreciprocal Cahn-Hilliard model.
+Then, all such nonreciprocal models studied in [15, 16, 47, 56]
+are recovered as special cases as is the complex Cahn-Hilliard
+equation appearing as a mass-conserving limiting case in
+Ref. [57].
+Before we embark on the derivation of the amplitude equa-
+tion we emphasize the wide spectrum of systems where the
+existence of two conservation laws allows for the occurrence
+of a conserved-Hopf instability and resulting intricate non-
+linear oscillatory behaviour.
+A prominent example is the
+spatio-temporal pattern formation of proteins vital for cellu-
+lar processes. Although chemical reactions cause conforma-
+tion changes of proteins, their overall number is conserved on
+the relevant time scale, e.g. MinE and MinD in ATP-driven
+cellular Min oscillations [19]. Other relevant examples in-
+clude oscillations in two-species chemotactic systems [54],
+an active poroelastic model for mechanochemical waves in
+cytoskeleton and cytosol [45], thin liquid layers covered by
+self-propelled surfactant particles [38, 43], oscillatory coupled
+lipid and protein dynamics in cell membranes [25], and heated
+two-layer liquid films [42] where the two interfaces may show
+intricate spatio-temporal oscillation patterns [39]. Normally,
+the conserved quantities in these examples correspond to con-
+centration fields, film thickness profiles, particle number den-
+sities and thus the conserved-Hopf instability may occur as a
+primary instability in the system. Another class of examples
+exists where it appears as a secondary instability. For exam-
+ple, in Marangoni convection the interaction between a large-
+scale deformational and a small-scale convective instability
+is described by coupled kinetic equations for the film height
+and a complex amplitude[18]. There, liquid layer profile and
+phase of the complex amplitude represent the two conserved
+quantities and the occurring conserved-Hopf instability corre-
+sponds to an oscillatory sideband instability. In all mentioned
+cases two conserved quantities are present in an sustained out-
+of-equilibrium setting.
+Systems like the given examples that feature two conserva-
+tion laws and are kept permanently out of equilibrium can be-
+come unstable through a conserved-Hopf instability, i.e., the
+oscillatory marginal linear mode (growth rate ∆(kc) = 0)
+occurs at zero wavenumber (kc = 0) and zero frequency
+(Ω(kc) = Ωc = 0). This is determined via a linear stabil-
+ity analysis of the trivial uniform steady state yielding the
+dispersion relations λ±(k) of the dominant pair of complex
+conjugate modes where ∆ = Re λ± and Ω = ±Im λ±. Al-
+though λ±(k = 0) = 0 always holds as the two conservation
+laws imply the existence of two neutral modes, the mode is
+nevertheless oscillatory at arbitrarily small wavenumbers, i.e.,
+directly beyond instability onset the system undergoes large-
+scale small-frequency oscillations. In other words, the conser-
+vation laws imply that above onset the first excited mode has
+the smallest wavenumber compatible with the domain bound-
+aries and oscillates on a correspondingly large time scale as
+Ω → 0 for k → 0. The weakly nonlinear behaviour is not
+covered by any of the seven amplitude equations summarized
+in section 1 of the Supplementary Material.
+Corresponding dispersion relations below, at and above in-
+0
+km
+k +
+k
+0
+∆m
+∆, Ω
+∼ ε
+∼ ε4
+FIG. 1. Dispersion relations λ±(k) below (blue line, δ < 0), at
+(purple line, δ = 0) and above (red line, δ > 0) the threshold of
+a conserved-Hopf instability as described by the series expansion
+Eq. (1). The solid lines represent the growth rate ∆ = Re λ± while
+the black dashed lines give the frequencies ±Ω = Im λ± that are
+identical in all three cases. Labeled thin dotted lines and solid bars
+indicate typical quantities and scalings above onset as described in
+the main text.
+stability threshold are sketched in Fig. 1 and are at small k
+given by
+λ±(k) =∆(k) ± iΩ(k)
+with ∆(k) =δk2 − ˜δk4 + O(k6)
+(1)
+and
+Ω(k) =ωk2 + ˜ωk4 + O(k6).
+The onset occurs when δ becomes positive while ˜δ > 0.
+Above onset, Eq. (1) indicates a band 0 < k < k+ =
+�
+δ/˜δ of
+exponentially growing wavenumbers with the fastest growth
+at km =
+�
+δ/(2˜δ) with rate ∆m = δ2/(4˜δ).
+To determine the universal kinetic equation describing the
+formation of spatio-temporal patterns in the vicinity of the on-
+set of a conserved-Hopf bifurcation with a dispersion relation
+as depicted in Fig. 1 we apply a weakly nonlinear approach
+[21]. First, we introduce a smallness parameter ε with |ε| ≪ 1
+and consider the system close to onset where δ = δ2ε2. From
+hereon subscript numerals indicate the order in ε of the cor-
+responding term. Then, the width of the band of growing
+wavenumbers and the maximal growth rate scale as ε and ε4,
+respectively. This determines the additional large spatial scale
+X = εx and slow timescale T = ε4t relevant for the dy-
+namics. Additionally, Eq. (1) indicates that the leading order
+oscillation frequency scales like Ω ≈ ωk2 ∼ ε2. This implies
+that a second slow timescale τ = ε2t has to be considered.
+Specifically, we now consider a homogeneous isotropic
+multi-component system with two conservation laws
+∂tρ = − ∇ · (Q(u)∇η(u))
+∂tσ = − ∇ · (R(u)∇µ(u))
+∂tn =F (u),
+(2)
+i.e., coupled kinetic equations for two conserved (ρ and σ)
+and N nonconserved [n = (n1, . . . , nN)] scalar field vari-
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+3
+ables. From here onwards, u = (ρ, σ, n) is used as abbrevi-
+ation where convenient. The dynamics of the two conserved
+quantities is given by the divergence of corresponding fluxes
+that consist of the product of a mobility (Q or R) and the gra-
+dient of a nonequilibrium (chemical) potential (η or µ) that
+may still depend on spatial derivatives. F represents a vec-
+tor of general functions of fields and their derivatives. The
+system may, for instance, represent a reaction diffusion sys-
+tem with N + 2 species that has been rearranged to explicitly
+show the two conservation laws or a thin-film description of
+an (N + 2)-component mixture where two components are
+nonvolatile. For an active system the potentials can not be
+obtained as variations from a single underlying energy func-
+tional. Here, we sketch the derivation of the amplitude equa-
+tions for a conserved-Hopf instability (Fig. 1) of a homoge-
+neous steady state while section 2 of the Supplementary Ma-
+terial present the complete algebra.
+To perform the weakly nonlinear analysis valid in the
+vicinity of instability onset, we expand all fields in ε, i.e.,
+u(X, τ, T) = u0 + εu1(X, τ, T) + ε2u2(X, τ, T) + . . . ,
+where u0 is the steady uniform state with F (u0) = 0 and
+ui(X, τ, T), i = 1, 2, . . . are the deviations that describe the
+(weakly) nonlinear behavior. We take account of the above
+discussed scaling of space and time implied by the dispersion
+relation by writing ∇x = ε∇X and ∂t = ε2∂τ + ε4∂T , re-
+spectively. With this we then Taylor-expand all mobilities and
+potentials in ε which allows us to consider Eqs. (2) order by
+order. The scaling implies that we need to successively con-
+sider all orders up to O(ε5) to discover evolution equations
+that capture dynamic effects on the slow timescale T.
+In principle, at each order we first determine the noncon-
+served fields as (nonlinear) functions of the conserved fields,
+reflecting that the dynamics of the former is slaved to the lat-
+ter. Second, we obtain the continuity equations to the cor-
+responding order by inserting the obtained expressions into
+the appropriate mobilities and potentials. In particular, at or-
+der ε, the contributions of the two continuity equations van-
+ish and the remaining N equations become a homogeneous
+linear algebraic system for the slaved quantities, solved by
+n1(X, τ, T) = nρρ1(X, τ, T) + nσσ1(X, τ, T) where nρ
+and nσ correspond to the zero eigenmodes (1, 0, nρ) and
+(0, 1, nσ) of the dominant eigenspace at k = 0. At order ε2,
+again the continuity equations are trivially fulfilled, and the
+remaining equations form an inhomogeneous linear algebraic
+system for the n2. Thereby, the inhomogeneity is nonlinear
+in lower order quantities. At order ε3, the first nonvanishing
+contributions from the continuity equations appear, that, after
+eliminating n1, correspond to linear equations for ρ1 and σ1.
+They provide the conditions for the instability onset at δ = 0
+in Eq. (1). They also capture the leading order oscillations
+with frequency ω on the time scale τ by an antisymmetric
+dynamic coupling that represents a nonreciprocal coupling of
+lowest order (a structure equivalent to the Schr¨odinger equa-
+tion for a free particle). Also for the n3 an inhomogeneous
+linear algebraic system emerges. At the subsequent order ε4,
+further contributions are obtained from the continuity equa-
+tions to the evolution on the time scale τ. Finally, at order
+ε5 we obtain expressions for ∂T ρ1 and ∂T σ1. Using the ear-
+lier obtained expressions for n1, n2, and n3, the complete
+continuity equations at this order can be written as nonlinear
+functions of the ρi and σi. This provides the weakly non-
+linear expression for the leading order time evolution on the
+timescale T. Next, the expressions found at the different or-
+ders are recombined, in passing “inverting” the scalings and
+Taylor expansions of time, coordinates and fields ρ and σ. The
+resulting system of two coupled amplitude equations is
+∂tϱ =∇
+��
+a0 + a1ϱ + a2ς + a3ϱ2 + a4ϱς + a5ς2�
+∇
+�
+α1ϱ + α2ς + α3ϱ2 + α4ϱς + α5ς2
++α6ϱ3 + α7ϱ2ς + α8ϱς2 + α9ς3 + α10∇2ϱ + α11∇2ς
+��
+,
+∂tς =∇
+��
+b0 + b1ϱ + b2ς + b3ϱ2 + b4ϱς + b5ς2�
+∇
+�
+β1ϱ + β2ς + β3ϱ2 + β4ϱς + β5ς2
++β6ϱ3 + β7ϱ2ς + β8ϱς2 + β9ς3 + β10∇2ϱ + β11∇2ς
+��
+,
+(3)
+where all coefficients are real and well defined through Tay-
+lor expansions of Q, R, η, µ, and F in Eqs. (2). The con-
+served fields ϱ(x, t) and ς(x, t) describe the spatial and tem-
+poral modulations in ρ(x, t) and σ(x, t) away from their re-
+spective mean values ρ0 and σ0. Eqs. (3) correspond to a
+general nonreciprocal two-component Cahn-Hilliard model.
+Introducing rescaled time, space, ϱ and ς one may set four
+parameters to specific values. In the particular case of con-
+stant mobilities (Q = Q0 and R = R0), as e.g. for stan-
+dard reaction-diffusion systems, one may also introduce al-
+ternative conserved fields in such a way that cross-couplings
+of highest order spatial derivatives are eliminated while the
+overall form of Eq. (3) is kept intact. Then, also the ad-hoc
+nonreciprocal Cahn-Hilliard models studied in [16, 47, 56]
+emerge as special cases. The corresponding parameter choices
+in Eq. (3) are given in Table III in Section 2 of the Supple-
+mentary Material. There, we also include two other limit-
+ing cases: (i) if certain symmetries between coefficients hold,
+one may introduce a complex amplitude A = ϱ + iς and
+present Eqs. (3) as a complex Cahn-Hilliard equation ∂tA =
+−G∇2 �
+ε + (1 + ib)∆ − (1 + ic)|A|2�
+A, i.e., as a complex
+Ginzburg-Landau equation with an additional outer Laplace
+operator reflecting the conservation property, as briefly con-
+sidered in Ref. [57]. This, in passing clarifies, that Eqs. (3)
+is more general than a “conserved complex Ginzburg-Landau
+equation” because it does not show its phase-shift invariance.
+(ii) Imposing another symmetry between coefficients renders
+the coupled equations variational.
+Then they correspond
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+4
+to amplitude equations for certain simultaneously occurring
+Cahn-Hilliard instabilities, as shown in Section 3 of the Sup-
+plementary Material, and represent a generic model for the
+dynamics of phase separation in a ternary system [31, 32].
+To conclude, we have derived the amplitude equation valid
+in the vicinity of a conserved-Hopf bifurcation - responsible
+for triggering qualitative transitions in a wide variety of out-
+of-equilibrium systems featuring two conservation laws. The
+derived model completes the set of universal amplitude equa-
+tions for the above discussed eight basic instabilities. Its im-
+portance equals that of the complex Ginzburg-Landau equa-
+tion that describes the universal behavior in the vicinity of
+a standard Hopf bifurcaton in systems without conservation
+laws [1, 11, 21, 41]. Note that the occurrence of additional
+subdominant neutral modes (e.g., resulting from additional
+conservation laws) or the simultaneous onset of several in-
+stabilities result in higher-codimension variants of the eight
+basic cases [12, 55]. Although it is known that the presented
+linear instability is a phenomenon that is not covered by the
+complex Ginzburg-Landau equation[38] only very few stud-
+ies have considered its (weakly) nonlinear behavior by corre-
+sponding amplitude equations [18, 40], normally, in special
+cases. On the one hand, Ref. [18] restricts its focus to am-
+plitude equations for spatially periodic traveling and standing
+waves, and on the other hand, Ref. [40] deals with a partic-
+ular case without reflection symmetry where one of the two
+conservation laws is weakly broken. The universal charac-
+ter of the model derived here, implies that literature results
+on the onset of motion and oscillations [16, 47, 56] as well
+as on the suppression of coarsening and the existence of lo-
+calized states [15, 16] are paradigmatic for the large class of
+out-of-equilibrium systems that undergo a conserved-Hopf in-
+stability and will, in consequence, represent universal features
+occurring in a wide variety of out-of-equilibrium systems with
+conservation laws.
+Acknowledgement TFH and UT acknowledge support from
+the doctoral school “Active living fluids” funded by the
+German-French University (Grant No.
+CDFA-01-14).
+In
+addition, TFH thanks the foundation “Studienstiftung des
+deutschen Volkes” for financial support.
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+
+6
+SUPPLEMENTARY MATERIAL
+1.
+Dispersion relations and amplitude equations
+This section of the Supplementary Material gives an overview of the eight types of linear instabilities of uniform steady states
+occurring in homogeneous isotropic systems decribed by scalar field variables. In particular, we discuss the corresponding
+dispersion relations and the seven well investigated amplitude equations (also called envelope equations) obtained by weakly
+nonlinear theory in the vicinity of the instability thresholds.
+The restriction to homogeneous isotropic systems implies that underlying model equations are translation- and rotation-
+invariant. For simplicity, in the following we only consider spatially one-dimensional systems, i.e., isotropy becomes reflec-
+tion symmetry (also called parity symmetry). The considered multi-component order parameter field u(x, t) represents a set of
+scalars. All occurring conservation laws are assumed to be local, i.e., the kinetic equation(s) for corresponding conserved fields
+ρ have the form of a continuity equation ∂tρ = −∂xj where j(x, t) is a flux that may depend linearly or nonlinearly on all
+components of u(x, t) and their spatial derivatives.
+Parity symmetry implies that each r.h.s. term features an even number of spatial derivatives. Here, x and t are position and
+time while ∂x and ∂t are the corresponding partial derivatives.
+The linear stability of steady uniform states u(x, t) = u0 is determined by adding small-amplitude perturbations that respect
+the boundary conditions and show an exponential time dependence eλt. In the case of infinitely extended translation-invariant
+systems, these are harmonics eikx with wavenumber k. Introducing u0 + χˆueλt+ikx with χ ≪ 1 into the kinetic equation
+and linearizing in χ gives the dispersion relation λ(k) and corresponding eigenmodes ˆu. The real part ∆ = Reλ is a rate that
+characterizes the exponential growth (∆ > 0) or decay (∆ < 0) of the corresponding linear mode. The imaginary part Ω := Imλ
+corresponds to a frequency that can be zero (stationary case) or nonzero (oscillatory case).
+nonconserved dynamics conserved dynamics
+homogeneous/large-scale, stationary
+Allen-Cahn (IIIs)
+Cahn-Hilliard (IIs)
+homogeneous/large-scale, oscillatory
+Hopfa (IIIo)
+conserved-Hopf (IIo)
+small-scale, stationary
+Turing (Is)
+conserved-Turing (-)
+small-scale, oscillatory
+waveb (Io)
+conserved-wave (-)
+a Also known as “Poincar´e-Andronov-Hopf”.
+b Also called “finite-wavelength Hopf”.
+TABLE I. Naming convention of linear instabilities (and corresponding bifurcations) classified via their spatial (homogeneous/large-scale vs.
+small-scale) and temporal (stationary vs. oscillatory) properties for the cases of nonconserved and conserved dynamics. In parentheses we give
+the names in the (incomplete) classification of Cross and Hohenberg [11].
+There are eight basic types of instability when basing the classification on the combination of three features: (i) large-scale vs.
+small-scale instability, (ii) stationary vs. oscillatory instability, and (iii) instability without and with conservation law(s). Each
+of them features typical dominant modes directly at and in the vicinity of the instability threshold. Using ε as control parameter,
+the left column of Fig. 2 presents the main types in our classification by showing the dispersion relation ∆(k). In each case we
+give relations below (ε < 0), at (ε = 0) and above (ε > 0) instability onset. We also indicate the critical wavenumber kc where
+at onset a maximum of ∆(k) touches zero, marginal wavenumber(s) k± where Reλ(k) crosses zero above onset, and the fastest
+growing wavenumber km where ∆(k) has a maximum above onset. Note for each of the four shown cases there exist a stationary
+and an oscillatory variant. The right column of Fig. 2 gives the loci of k± and km in the (k, ε)-plane thereby illustrating the
+band of unstable wavenumbers in its dependence on ε. Note that these dependencies are based on the leading order amplitude
+equation in each case. When higher orders are included k±(ε) and km(ε) can be quantitatively different for ε ̸= 0. Our naming
+convention for the eight instabilities is given in Table I. For reference, the classification of Cross and Hohenberg [11] is given,
+but note that it only distinguishes six cases. Ref. [11] further states that “type II can often be scaled to resemble type I”. In
+our opinion this is not correct. Also, their statements that on the one hand stationary and oscillatory instabilities have at onset
+frequencies Ω = 0 and Ω = O(1), respectively, and on the other hand that type II occurs in the presence of a conservation law
+seem to contradict each other, as the oscillatory case II has Ω ∼ O(ε2). In contrast, we propose a classification that takes the
+importance of conservation laws directly and systematically into account.
+Inspecting the first and third row of Fig. 2 we see that there are four basic instability types for nonconserved systems: If
+the critical wavenumber, i.e., the marginal wavenumber at onset is zero (kc = 0) the marginal mode is homogeneous (some-
+times called global) as it synchronously affects each point of the domain without any spatial modulation. We refer to it as
+a homogeneous, uniform or global instability. If the corresponding critical frequency Ωc is zero [nonzero] the instability is
+stationary [oscillatory]. The corresponding amplitude equations for the stationary (Allen-Cahn instability) and the oscillatory
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+7
+0
+1
+2
+k
+1
+0
+1
+∆
+ε > 0
+ε = 0
+ε < 0
+km = kc = 0
+k +
+(a)
+0
+1
+2
+k
+0
+1
+2
+ε
+km
+unstable
+k +
+(b)
+0
+1
+2
+k
+1
+0
+1
+∆
+ε > 0
+ε = 0
+ε < 0
+km
+kc
+kc = 0
+k +
+(c)
+0
+1
+2
+k
+0
+1
+2
+ε
+km
+unstable
+k +
+(d)
+0
+1
+2
+k
+1
+0
+1
+∆
+ε > 0
+ε = 0
+ε < 0
+km = kc
+0
+k +
+k −
+(e)
+0
+1
+2
+k
+0
+1
+2
+ε
+km
+unstable
+k −
+k +
+(f)
+0
+1
+2
+k
+1
+0
+1
+∆
+ε > 0
+ε = 0
+ε < 0
+kc
+0
+k3.5
+kc
+k +
+k −
+(g)
+0
+1
+2
+k
+0
+1
+2
+ε
+km
+unstable
+k −
+k +
+(h)
+FIG. 2. Classification of dispersion relations distinguishing large-scale and small-scale instabilities as well as instabilities without and with
+conservation law(s). Shown are (left) the real part ∆ = Re λ(k) of the dispersion relation and (right) the position of marginal wavenumber(s)
+k± (solid line) and fastest growing wavenumber km (dashed line) in the plane spanned by wavenumber k and control parameter ε. For each
+of the four cases there exist stationary and oscillatory variants. In the left panels we also indicate the critical wavenumber kc where the
+instability onset occurs (at ε = 0, heavy solid blue line). Dashed lines give ∆(k) below (ε < 0) and above (ε > 0) onset. Shown are (a,b) the
+nonconserved homogeneous (Allen-Cahn and Hopf) instability, (c,d) the conserved large-scale (Cahn-Hilliard and conserved-Hopf) instability,
+(e,f) the nonconserved small-scale (Turing and wave) instability, and (g,h) the conserved small-scale (conserved-Turing and conserved-wave)
+instability. Naming conventions are summarizd in Table I.
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+8
+(Hopf instability) case are the Allen-Cahn equation
+∂tB = sgn(ε)B + ∂xxB + αB2 − B3
+(4)
+and the complex Ginzburg-Landau equation
+∂tA = sgn(ε)A + (1 + iκi)∂xxA − (1 + iαi)|A|2A
+(5)
+respectively.
+Here, B(x, t) is the spatially slowly varying real amplitude of the spatially homogeneous stationary mode while A(x, t) is the
+spatially slowly varying complex amplitude of the spatially homogeneous oscillatory mode at k = 0. Note that a derivation of the
+universal amplitude equations results in specific coefficients for each term on the right hand side that depend on the corresponding
+original model. Furthermore, the space and time variables describe spatially and temporally slow dynamics, hence, amplitude
+equations are often referred to as envelope equations. Introducing a transformation of space, time and amplitude one can always
+eliminate three coefficients. Here and for the following cases we give these simplified universal equations in the supercritical
+cases. In the corresponding subcritical cases the cubic nonlinearity acts destabilizing, e.g. it occurs with a positive sign in Eq. (5).
+Then, higher order stabilizing terms, e.g. a quintic nonlinearity, must be included to obtain a well behaved system. The sgn(ε)
+function occurs since it determines whether the amplitude linearly grows or decays.
+In contrast, a nonzero marginal wavenumber at onset (kc ̸= 0) indicates a small-scale instability, i.e., an instability of finite
+wavelength. The amplitude equation in the stationary case (Turing instability) is the real Ginzburg-Landau equation
+∂tA = sgn(ε)A + ∂xxA − |A|2A
+(6)
+where A(x, t) is the slowly varying complex amplitude of the spatially periodic stationary mode at k = kc. In the oscillatory
+case (wave instability) the coupled complex Ginzburg-Landau equations
+∂tA1 = sgn(ε)A1 − c∂xA1 + (1 + iκi)∂xxA1 − (1 + iαi)|A1|2A1 − (βr + iβi)|A2|2A1
+∂tA2 = sgn(ε)A2 + c∂xA2 + (1 + iκi)∂xxA2 − (1 + iαi)|A2|2A2 − (βr + iβi)|A1|2A2
+(7)
+emerge as amplitude equations where complex amplitudes A1 and A2 correspond to left and right traveling waves, respectively.
+These equations are only valid for small group velocity c (see sec. VI.E of [1], section 7.1 of [21], sections 1 & 2 of [53]). If
+this condition is not fulfilled a nonlocal equation is derived [26]. The four described cases for systems without conservation
+laws are all well covered by the Cross-Hohenberg classification (as types Is, Io, IIIs, IIIo) [11] and are widely analyzed in the
+literature.
+However, for systems with a conservation law there are another four basic types again distinguished based on wavenumber
+and mode type at onset. They are shown in the second and fourth row of Fig. 2. The conservation law results in the existence of a
+neutral mode at zero wavenumber, i.e., λ = 0 at k = 0 at all values of ε. Note that the instability thresholds are equivalent to the
+corresponding cases without conservation law. However, the fastest growing mode above onset behaves differently. For instance,
+in the case of zero marginal wavenumber at onset (kc = 0, second row of Fig. 2, Cahn-Hilliard instability), km increases with
+ε in contrast to the case of an Allen-Cahn instability. This implies that the instability is observed as a large-scale instability, not
+a homogeneous one. The growth of a homogeneous mode is incompatible with a fully conserved dynamics. The corresponding
+amplitude equation has only recently been systematically derived [2]. It corresponds to the Cahn-Hilliard equation (well known
+from other contexts)
+∂tB = ∂xx
+�
+−sgn(ε)B + αB2 + B3 − ∂xxB
+�
+(8)
+where B(x, t) is the spatially slowly varying real amplitude of the spatially homogeneous stationary mode at k = 0. Since the
+Cahn-Hilliard equation is a continuity equation the total growth, i.e., the growth integrated over the domain,
+�
+dx∂tB, vanishes.
+This confirms that conserved quantities can only be spatially redistributed within the domain but overall do not change.
+The small-scale cases with conservation law [Fig. 2 (g,h)] are not explicitely mentioned in the Cross-Hohenberg classification
+[11], but their relevance and distinct features became later more widely known [34, 53]. Here, we call the stationary and
+oscillatory case conserved-Turing and conserved-wave instability, respectively. The corresponding amplitude equations in the
+stationary case correspond to a real Ginzburg-Landau equation coupled to a nonlinear diffusion equation (see [34] and section 9.4
+of [21])
+∂tA = sgn(ε)A + ∂xxA − |A|2A − AB
+∂tB = ∂xx
+�
+νB + µ|A|2�
+.
+(9)
+Here, A is the slowly varying complex amplitude of a spatially periodic stationary mode and B is the real amplitude of a
+spatially slowly varying stationary mode. In the oscillatory case one finds two complex Ginzburg-Landau equations (for complex
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+9
+amplitudes A1 and A2 of left and right traveling waves, respectively) coupled to an equation for a real scalar mode of amplitude
+B (see pg. 1034 of [53]), i.e.,
+∂tA1 = sgn(ε)A1 − c∂xA1 + (1 + iκi)∂xxA1 − (1 + iαi)|A1|2A1 − (βr + iβi)|A2|2A1 − (1 + iγi)A1B
+∂tA2 = sgn(ε)A2 + c∂xA2 + (1 + iκi)∂xxA2 − (1 + iαi)|A2|2A2 − (βr + iβi)|A1|2A2 − (1 + iγi)A2B
+∂tB = ∂xx
+�
+νB + µ(|A1|2 − |A2|2)
+�
+.
+(10)
+Again, these are only valid for small group velocity c.
+Although the presented picture might at first sight seem complete, the careful reader will have noticed that Eqs. (4) to (10) only
+present the amplitude equations for seven of the eight discussed cases of linear instabilities: we have neglected the large-scale
+oscillatory instability with conservation law (conserved-Hopf instability). This case is to our knowledge not yet systematically
+treated in the literature, and the corresponding universal amplitude equation has not yet been systematically derived. This is the
+subject of the main text.
+2.
+Derivation of amplitude equation for conserved-Hopf instability
+To provide the detailed derivation of the general nonreciprocal Cahn-Hilliard equation as amplitude equation for the
+conserved-Hopf instability, i.e., a large-scale oscillatory instability in systems with conservation law, we consider a reflection-
+symmetric multi-component system. It is written as coupled kinetic equations for two conserved (ρ and σ) and N nonconserved
+(n = (n1, . . . , nN)) scalar field variables
+∂tρ = − ∂x (Q(u)∂xη(u))
+∂tσ = − ∂x (R(u)∂xµ(u))
+∂tn =F (u).
+(11)
+Form here onwards u = (ρ, σ, n) is used as abbreviation where convenient. The dynamics of the conserved quantities is
+determined by the divergence of the corresponding fluxes. Each flux is the product of a mobility function (Q and R, respectively)
+and the gradient of a potential (η and µ, respectively). In general, these are nonequilibrium (chemical) potentials that can not be
+obtained as variations from a single underlying energy functional. This renders the system active. For scalar fields, reflection
+symmetry implies that Eqs. (11) are invariant under the transformation x → −x. Therefore, the individual terms within the
+potentials η and µ do either not include derivatives or an even number of them. Normally, the mobilities are assumed to be
+functions of the fields, but terms with an even number of derivatives may also be easily accommodated. Furthermore, at least
+one uniform steady state u0 with F (u0) = 0 shall exist.
+We consider the case of a conserved-Hopf instability, i.e., at control parameter δ = 0 the considered u0 state becomes unstable
+with a dispersion relation above onset as in Fig. 1 of the main text. Note that in a system with N ≥ 2 nonconserved quantities
+a standard Hopf instability is also still possible. However, then the conserved quantities do not contribute to the corresponding
+linear modes, and interactions between nonconserved oscillatory and conserved real modes will only occur nonlinearly. Here,
+such a setting is not considered. The conserved-Hopf instability we are here interested in always involves the branches of the
+dispersion relation containing the two neutral modes at k = 0.
+To perform the weakly nonlinear analysis valid in the vicinity of instability onset, i.e., for δ = δ2ε2, we expand all fields in
+the smallness parameter ε ≪ 1, i.e.,
+u(X, τ, T) = u0 + εu1(X, τ, T) + ε2u2(X, τ, T) + . . . ,
+(12)
+take account of the scaling discussed in the main text by writing ∂x = ε∂X and ∂t = ε2∂τ +ε4∂T , and also expand all quantities
+occurring on the right hand side of Eqs. (11) as
+Q =Q0 + εQ1 + ε2Q2 + . . .
+R =R0 + εR1 + ε2R2 + . . .
+η =η0 + εη1 + ε2η2 + ε3η3 + . . .
+µ =µ0 + εµ1 + ε2µ2 + ε3µ3 + . . .
+F =εF 1 + ε2F 2 + ε3F 3 + . . . ,
+(13)
+where F 0 = F (u0) = 0 was used. The various coefficients are given by corresponding Taylor expansions, e.g. Q2 = u1 ·
+1
+2
+∂2Q
+∂u2
+��
+u0 · u1 + ∂Q
+∂u
+��
+u0 · u2. Examples for these coefficients are given in Table II. All expansions and scalings are inserted into
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+10
+the model equations (11), that are then considered order by order in ε. Due to the scaling implied by the dispersion relation, one
+needs to successively consider all orders up to O(ε5) to obtain evolution equations that capture dynamics effects on the slow
+timescale T.
+The general procedure to follow at each order i = 1, . . . , 5 is: First, determine the nonconserved fields ni as (nonlinear)
+functions of the conserved fields σ and ρ, hence, consider the dynamics of the nonconserved fields that is slaved to the dynamics
+of the conserved fields. Second, obtain the continuity equations to the corresponding order by inserting the expressions for ni
+into the appropriate mobilities Qi, Ri and potentials ηi, µi. Finally, the dynamics for ρ and σ are combined into two coupled
+amplitude equations including terms up to ∂T ρ1 and ∂T σ1, respectively. Now we proceed order by order.
+a.
+Order ε:
+As expected, we recover the linear result at k = 0: The contributions of the two continuity equations vanish
+and the remaining N equations become the algebraic system
+0 = F 1
+(14)
+linear in N + 2 unknown quantities u1. It is solved for
+n1(X, τ, T) = nρρ1(X, τ, T) + nσσ1(X, τ, T)
+(15)
+where nρ and nσ correspond to the zero eigenmodes (1, 0, nρ) and (0, 1, nσ), respectively, discussed in the main text.
+b.
+Order ε2:
+As in O(ε), the continuity equations are trivially fulfilled and the nonconserved fields give
+0 = F 2 .
+(16)
+These algebraic relations consist of a linear part similar to Eq. (15) but for the fields u2 and a nonlinear part quadratic in ρ1
+and σ1 (after eliminating quadratic parts involving n1 via Eq. (15)). The nonlinearities correspond to the inhomogeneity of the
+algebraic system for the u2. In consequence, n2 is now given as a sum of a part linear in the amplitudes ρ2 and η2 and the
+nonlinearity, namely,
+n2 = nρρ2 + nσσ2 + nρρρ2
+1 + nρσρ1σ1 + nσσσ2
+1 .
+(17)
+Here and in the following, the vectors of real constant coefficients nα, nαβ, etc. depend on the specific functions F .
+c.
+Order ε3:
+At the next order, the first nonvanishing contribution from the continuity equations appears, resulting in the
+system
+∂τρ1 = −∂X (Q0∂Xη1)
+∂τσ1 = −∂X (R0∂Xµ1)
+∂τn1 = F 3
+(18)
+where η1 and µ1 are linear in ρ1, σ1 and n1. For the latter we insert Eq. (15) and obtain
+η1 = ηρρ1 + ησσ1
+µ1 = µρρ1 + µσσ1
+(19)
+with real constant coefficients ηρ, ησ, µρ and µσ as exemplified in Table II that also provides further coefficients appearing at
+higher orders in ε. Further, Q0, R0 are constants and inserting the expressions (19) we can write the first two equations in (18)
+as a linear system
+∂τ
+�
+ρ1
+σ1
+�
+= −
+�
+Q0ηρ Q0ησ
+R0µρ R0µσ
+�
+∂XX
+�
+ρ1
+σ1
+�
+.
+(20)
+Applying (ρ1, σ1) ∼ exp (ikX + λτ), its eigenvalues are
+λ± = k2 Q0ηρ + R0µσ
+2
+± k2
+�
+(Q0ηρ − R0µσ)2
+4
++ Q0ησR0µρ .
+(21)
+Comparing to the dispersion relation (Eq. (1) of the main text) allows us to identify
+δ =Q0ηρ + R0µσ
+2
+iω =
+�
+(Q0ηρ − R0µσ)2
+4
++ Q0ησR0µρ .
+(22)
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+11
+At onset of the conserved-Hopf instability the growth rate δ vanishes, i.e.,
+Q0ηρ = −R0µσ
+(23)
+and the eigenvalues are purely imaginary, i.e.,
+(Q0ηρ − R0µσ)2
+4
++ Q0ησR0µρ < 0 .
+(24)
+Using (23), this implies
+ησµρ < ηρµσ < 0,
+(25)
+and thereby defines a nonreciprocity condition for the linear coupling terms within the potentials. Since we are interested in the
+dynamics closely above [below] the instability onset where δ = δ2ε2 with δ2 > 0 [δ2 < 0], Eq. (23) only holds at leading order,
+i.e., including the next order we have Q0ηρ = −R0µσ +(2δ2 −R0µσ)ε2. This makes the result fully consistent with the scaling
+based on Eqs. (1), as the oscillations of leading order frequency ωk2 occur on the timescale τ [given that ω = O(1)] where the
+growth rate vanishes. Growth only occurs on the slower time scale T. Specifically, the O(ε2) contribution in δ is then considered
+when below discussing order ε5 terms. Here, at O(ε3), we can simply set δ = 0. With this, we now reformulate Eq. (20). Note
+that any linear combination of ρ and σ, e.g.
+A(X, τ, T) = aρρ(X, τ, T) + aσσ(X, τ, T)
+(26)
+B(X, τ, T) = bρρ(X, τ, T) + bσσ(X, τ, T) .
+(27)
+with coefficients aρ, aσ, bρ and bσ results in an equivalent formulation for alternative conserved fields A and B. We use the
+resulting freedom to simplify Eq. (20) by choosing
+bσ = −ωaρ + Q0ηρbρ
+R0µρ
+,
+aσ = ωbρ − Q0ηρaρ
+R0µρ
+(28)
+and aρ, bρ are normalization constants. This reduces Eqs. (20) to
+∂τ
+�
+A1
+B1
+�
+=
+�
+0 −ω
+ω
+0
+�
+∂XX
+�
+A1
+B1
+�
+,
+(29)
+i.e., the leading order oscillation is represented by an antisymmetric dynamic coupling of A and B, that represents the lowest
+order nonreciprocal coupling. The form of Eq. (29) allows one to easily show that the corresponding part of the dynamics is not
+dissipative.
+For simplicity of notation, here, we proceed with the description using the original quantities ρ and σ. It helps us to identify
+the structure of the two continuity equations in terms of Q, R, η and µ. Still at O(ε3), we determine n3 via the third equation in
+(18): we insert n1 from Eq. (15) and use Eqs. (20) for the time derivative to obtain
+n3 =nρρ3 + nσσ3 + nρρ2ρ1ρ2 + nρσ (ρ1σ2 + ρ2σ1) + nσσ2σ1σ2
++ nρρρρ3
+1 + nρρσρ2
+1σ1 + nρσσρ1σ2
+1 + nσσσσ3
+1
++ nρxx∂XXρ1 + nσxx∂XXσ1 .
+(30)
+Similar to the expression for n2 it consists of the sum of a linear part given by Eq. (15) applied to u3 and a nonlinear part that
+corresponds to an inhomogeneity.
+d.
+Order ε4:
+At the next order we obtain
+∂τρ2 = −∂X (Q0∂Xη2) − ∂X (Q1∂Xη1)
+∂τσ2 = −∂X (R0∂Xµ2) − ∂X (R1∂Xµ1)
+∂τn2 = F 4
+(31)
+where additionally to the already known Q0, R0, η1 and µ1 the higher order quantities Q1, R1, η2 and µ2 enter. Using Eq. (17)
+for n2 we obtain
+Q1 =qρρ1 + qσσ1
+R1 =rρρ1 + rσσ1
+η2 =ηρρ2 + ησσ2 + ηρρρ2
+1 + ηρσρ1σ1 + ησσσ2
+1
+µ2 =µρρ2 + µσσ2 + µρρρ2
+1 + µρσρ1σ1 + µσσσ2
+1
+(32)
+Furthermore, at order ε4 one may also determine an algebraic expression for n4. However, here, we do not present it because
+it does not contribute to the relevant leading order dynamics of ρ and σ.
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+12
+e.
+Order ε5:
+Finally, the fifth order gives the kinetic equations
+∂T ρ1 + ∂τρ3 = −∂X (Q0∂Xη3) − ∂X (Q1∂Xη2) − ∂X (Q2∂Xη1)
+∂T σ1 + ∂τσ3 = −∂X (R0∂Xµ3) − ∂X (R1∂Xµ2) − ∂X (R2∂Xµ1)
+∂T n1 + ∂τn3 = F 5 .
+(33)
+Using the already determined expressions for n1, n2 and n3 the complete right hand sides of the continuity Eqs. (33) can be
+written as nonlinear functions of the ρi and σi with coefficients defined in a similar way as at lower orders:
+Q2 =qρρ2 + qσσ2 + qρρρ2
+1 + qρσρ1σ1 + qσσσ2
+1
+R2 =rρρ2 + rσσ2 + rρρρ2
+1 + rρσρ1σ1 + rσσσ2
+1
+η3 =ηρρ3 + ησσ3 + 2ηρρρ1ρ2 + ηρσ (ρ1σ2 + ρ2σ1) + 2ησσσ1σ2
++ ηρρρρ3
+1 + ηρρσρ2
+1σ1 + ηρσσρ1σ2
+1 + ησσσσ3
+1 + ηρxx∂XXρ1 + ησxx∂XXσ1
+µ3 =µρρ3 + µσσ3 + 2µρρρ1ρ2 + µρσ (ρ1σ2 + ρ2σ1) + 2µσσσ1σ2
++ µρρρρ3
+1 + µρρσρ2
+1σ1 + µρσσρ1σ2
+1 + µσσσσ3
+1 + µρxx∂XXρ1 + µσxx∂XXσ1
+(34)
+This provides the weakly nonlinear expression for the time evolution on the timescale T. To obtain the final amplitude
+equations we combine the dynamics found at the different orders. In other words, we recombine the different orders of the
+expansion of the fields into the appropriate fields u as the deviations from the steady state u0. Specifically we introduce
+ϱ ≡ ρ − ρ0 and ς ≡ σ − σ0.
+For instance, the dynamics of ϱ is given by
+∂tϱ =ε3∂τρ1 + ε4∂τρ2 + ε5 (∂T ρ1 + ∂τρ3) + O(ε6)
+= − ε3∂X (Q0∂Xη1) − ε4∂X (Q0∂Xη2 + Q1∂Xη1)
+− ε5∂X (Q0∂Xη3 + Q1∂Xη2 + Q2∂Xη1) + O(ε6)
+= − ε2∂X
+��
+Q0 + εQ1 + ε2Q2
+�
+∂X
+�
+εη1 + ε2η2 + ε3η3
+��
++ O(ε6).
+(35)
+Note that in the last step we have added selected terms, e.g. −ε6∂X (Q2∂Xη2) that would naturally occur at higher orders of the
+derivation. However, including them in Eq. (35) obtained through considerations up to O(ε5) allows us to conserve the structure
+of a continuity equation (even with a flux that equals the product of a mobility and a gradient of a potential). We emphasize that
+this procedure does not correspond to a further approximation. The leading order terms, i.e., terms up to O(ε5) are not touched
+and adding terms that are smaller than O(ε5) does not change the validity of Eqs. (35). One may even argue that one has to
+add these terms to keep the structure as a conservation law intact. See the corresponding discussion for the related problem of a
+gradient dynamics structure in appendix A of [49].
+Furthermore, we introduce the original scales, x and t and the fields ϱ and ς, e.g.
+ηρρε2∂XX
+�
+ε2ρ2
+1 + 2ε3ρ1ρ2
+�
+= ηρρε2∂XX
+�
+ερ1 + ε2ρ2
+�2 + O(ε6) = ηρρ∂xxϱ2 + O(ε6),
+(36)
+and obtain as result the coupled amplitude equations
+∂tϱ = − ∂x
+��
+Q0 + qρϱ + qσς + qρρϱ2 + qρσϱς + qσσς2�
+∂x
+�
+ηρϱ + ησς + ηρρϱ2 + ηρσϱς + ησσς2
++ηρρρϱ3 + ηρρσϱ2ς + ηρσσϱς2 + ησσσς3 + ηρxx∂xxϱ + ησxx∂xxς
+��
+∂tς = − ∂x
+��
+R0 + rρϱ + rσς + rρρϱ2 + rρσϱς + rσσς2�
+∂x
+�
+µρϱ + µσς + µρρϱ2 + µρσϱς + µσσς2
++µρρρϱ3 + µρρσϱ2ς + µρσσϱς2 + µσσσς3 + µρxx∂xxϱ + µσxx∂xxς
+��
+(37)
+where all coefficients are real and well defined through Taylor expansions (see Table II). By construction the mean values of ϱ and
+ς vanish, i.e.,
+�
+dV ϱ =
+�
+dV ς = 0. For the sake of readability we use another naming convention for the parameters in Eqs. (3)
+of the main text where we incorporate the overall minus sign into the coefficients of the terms in the potentials. Eqs. (37)
+correspond to a general nonreciprocal Cahn-Hilliard model. Note that all nonreciprocal Cahn-Hilliard models studied in the
+literature [16, 47, 56, 57] correspond to particular choices of relations between parameters in the derived general nonreciprocal
+Cahn-Hilliard model (37). These choices are listed in Table III.
+In matrix form Eqs. (37) read
+∂t
+�
+ϱ
+ς
+�
+= −∂x
+�
+M(ϱ, ς)∂x
+�
+L
+�
+ϱ
+ς
+�
++
+�
+Nρ(ϱ, ς)
+Nσ(ϱ, ς)
+�
++ D∂xx
+�
+ϱ
+ς
+���
+(38)
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+13
+where M(ϱ, ς) is a non-constant diagonal mobility matrix, and L and D are fully occupied constant matrices that describe the
+terms within the potentials that are linear in the fields and linear in their second spatial derivatives, respectively. All nonlinearities
+within the potentials are contained in the Nρ(ϱ, ς) and Nσ(ϱ, ς). If we linearize Eqs. (38) we obtain
+∂t
+�
+ϱ
+ς
+�
+= − ∂x
+�
+M(0, 0)∂x
+�
+L
+�
+ϱ
+ς
+�
++ D∂xx
+�
+ϱ
+ς
+���
+⇒ ∂t
+�
+ϱ
+ς
+�
+= −
+�
+Q0ηρ Q0ησ
+R0µρ R0µσ
+�
+∂xx
+�
+ϱ
+ς
+�
+−
+�
+Q0ηρxx Q0ησxx
+R0µρxx R0µσxx
+�
+∂xxxx
+�
+ϱ
+ς
+�
+.
+(39)
+Determining the eigenvalues we recover the original dispersion relation (1) as it should be. In particular, the coefficients in
+Eq. (39) are related to the ones of Eq. (1) of the main text, i.e.,
+˜δ = − Q0ηρxx + R0µσxx
+2
+(40)
+i˜ω =ηρηρxxQ2
+0 + (2ησxxµρ + 2ησµρxx − ηρxxµσ − ηρµσxx)Q0R0 + µσµσxxR2
+0
+2
+�
+η2ρQ2
+0 − 2ηρµσQ0R0 + R0(4ησµρQ0 + µ2σR0)
+(41)
+together with the already known relations for δ2 and ω from Eqs. (22). Note that in contrast to the dynamics on the timescale τ,
+here, we take the deviation from the onset of instability into account, i.e., (Q0ηρ + R0µσ)/2 = δ2ε2.
+Finally, we may remove the cross-couplings in the highest order derivatives by formulating Eqs. (38) in alternative conserved
+amplitudes
+�
+A
+B
+�
+≡ T
+�
+ϱ
+ς
+�
+where T is the corresponding transformation matrix. Multiplying Eqs. (38) with T we rewrite
+it in the new field variables as
+∂t
+�
+A
+B
+�
+= −∂x
+�
+T �
+M(A, B)T−1∂x
+�
+T L T−1
+�
+A
+B
+�
++ T
+�
+�
+Nρ(A, B)
+�
+Nσ(A, B)
+�
++ T D T−1∂xx
+�
+A
+B
+���
+(42)
+where the quantities with tilde are obtained by e.g. M(ϱ, ς) = M(ϱ(A, B), ς(A, B)) = �
+M(A, B). If we pick a transformation
+matrix T such that T D T−1 is diagonal the fourth order derivative cross-coupling terms are eliminated. However, in general,
+this also renders the resulting mobility matrix T �
+M T−1 nondiagonal. In contrast, if the mobility matrix is constant, i.e.,
+M = M0, it can be moved behind the second spatial derivative. Then, one can pick a matrix T that diagonalises the product
+M0 D and eliminate the fourth order spatial derivative cross-coupling within the potentials without re-introducing it through
+dynamic cross-coupling terms.
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+14
+Quantities in
+Eq. (13)
+Q0 = Q(u0)
+Q1 =
+�
+i
+∂Q
+∂ui
+����
+u0
+(u1)i
+Q2 = 1
+2
+�
+i,j
+∂2Q
+∂ui∂uj
+����
+u0
+(u1)i (u1)j +
+�
+i
+∂Q
+∂ui
+����
+u0
+(u2)i
+Rm, ηm, µm, F m for m = 0, 1, 2 analogously
+η3 = 1
+6
+�
+i,j,k
+∂3η
+∂ui∂uj∂uk
+����
+u0
+(u1)i (u1)j (u1)k +
+�
+i
+∂η
+∂ (∆ui)
+����
+u0
+(∆u1)i
++
+�
+i,j
+∂2η
+∂ui∂uj
+����
+u0
+(u1)i (u2)j +
+�
+i
+∂η
+∂ui
+����
+u0
+(u3)i
+µ3, F 3
+analogously
+Eq. (19)
+ηρ = ∂η
+∂ρ
+����
+u0
++
+�
+i
+∂η
+∂ni
+����
+u0
+(nρ)i
+ησ = ∂η
+∂σ
+����
+u0
++
+�
+i
+∂η
+∂ni
+����
+u0
+(nσ)i
+µρ, µσ
+analogously
+Eq. (32)
+qρ = ∂Q
+∂ρ
+����
+u0
++
+�
+i
+∂Q
+∂ni
+����
+u0
+(nρ)i
+qσ = ∂Q
+∂σ
+����
+u0
++
+�
+i
+∂Q
+∂ni
+����
+u0
+(nσ)i
+rρ, rσ
+analogously
+ηρρ =1
+2
+∂2η
+∂ρ2
+����
+u0
++
+�
+i
+∂2η
+∂ρ∂ni
+����
+u0
+(nρ)i + 1
+2
+�
+i,j
+∂2η
+∂ni∂nj
+����
+u0
+(nρ)i (nρ)j
+ηρσ = ∂2η
+∂ρ∂σ
+����
+u0
++
+�
+i
+∂2η
+∂ρ∂ni
+����
+u0
+(nσ)i +
+�
+i
+∂2η
+∂σ∂ni
+����
+u0
+(nρ)i +
+�
+i,j
+∂2η
+∂ni∂nj
+����
+u0
+(nρ)i (nσ)j
+ησσ =1
+2
+∂2η
+∂σ2
+����
+u0
++
+�
+i
+∂2η
+∂σ∂ni
+����
+u0
+(nσ)i + 1
+2
+�
+i,j
+∂2η
+∂ni∂nj
+����
+u0
+(nσ)i (nσ)j
+µρρ, µρσ, µσσ
+analogously
+Eq. (34)
+qρρ =1
+2
+∂2Q
+∂ρ2
+����
+u0
++
+�
+i
+∂2Q
+∂ρ∂ni
+����
+u0
+(nρ)i + 1
+2
+�
+i,j
+∂2Q
+∂ni∂nj
+����
+u0
+(nρ)i (nρ)j
+qρσ, qσσ, rρρ, rρσ, rσσ
+analogously
+ηρρρ =1
+6
+∂3η
+∂ρ3
+����
+u0
++ 1
+2
+�
+i
+∂3η
+∂ρ2∂ni
+����
+u0
+(nρ)i + 1
+2
+�
+i,j
+∂3η
+∂ρ∂ni∂nj
+����
+u0
+(nρ)i (nρ)j + 1
+6
+�
+i,j,k
+∂3η
+∂ni∂nj∂nk
+����
+u0
+(nρ)i (nρ)j (nρ)k
+ηρρσ =1
+2
+∂3η
+∂ρ2∂σ
+����
+u0
++ 1
+2
+�
+i
+∂3η
+∂ρ2∂ni
+����
+u0
+(nσ)i +
+�
+i
+∂3η
+∂ρ∂σ∂ni
+����
+u0
+(nρ)i +
+�
+i,j
+∂3η
+∂ρ∂ni∂nj
+����
+u0
+(nρ)i (nσ)j
++ 1
+2
+�
+i,j
+∂3η
+∂σ∂ni∂nj
+����
+u0
+(nρ)i (nρ)j + 1
+2
+�
+i,j,k
+∂3η
+∂ni∂nj∂nk
+����
+u0
+(nρ)i (nρ)j (nσ)k
+ηρσσ, ησσσ, µρρρ, µρρσ, µρσσ, µσσσ
+analogously
+ηρxx =
+∂η
+∂ (∆ρ)
+����
+u0
++
+�
+i
+∂η
+∂ (∆ni)
+����
+u0
+(nρ)i
+ησxx, µρxx, µσxx analogously
+TABLE II. Relations between the quantities in the original equation (11) and the various coefficients appearing in equations (13), (19), (32),
+and (34) at the different orders in ε.
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+15
+Model
+field variables
+zero coefficients
+nonzero coefficients
+You et al. [56] and Frohoff-H¨ulsmann et al. [16]
+∂tφµ = ∆
+�
+−γµ∆φµ + χµφµ + 1
+3φ3
+µ + κµνφν
+�
+with
+1
+|V |
+�
+V dV φµ = ¯φ0
+µ
+µ = A, B , ν ̸= µ
+ϱ = φA − φ0
+A
+ς = φB − φ0
+B
+ηρσ, ησσ
+µρρ, µρσ
+ηρρσ, ηρσσ, ησσσ
+µρρρ, µρρσ , µρσσ
+ησxx
+µρxx
+ηρ = χA +
+�
+φ0
+A
+�2
+µσ = χB +
+�
+φ0
+B
+�2
+ησ = κAB
+µρ = κBA
+ηρρ = φ0
+A
+µσσ = φ0
+B
+ηρρρ = µσσσ = 1
+3
+ηρxx = −γA
+µσxx = −γB
+Saha et al. [47]
+∂tφ1 = ∆
+�
+− κ∆φ1 + 2(c1,1 + c1,2)2φ1 − 6(c1,1 + c1,2)φ2
+1
++4φ3
+1 + (χ + α)φ2 + 2χ′φ1φ2
+2
+�
+∂tφ2 = ∆
+�
+− κ∆φ2 + 2(c2,1 + c2,2)2φ2 − 6(c2,1 + c2,2)φ2
+2
++4φ3
+2 + (χ − α)φ1 + 2χ′φ2φ2
+1
+�
+with
+1
+|V |
+�
+V dV φi = ¯φi
+i = 1, 2
+ϱ = φ1 − ¯φ1
+ς = φ2 − ¯φ2
+ηρρσ, ησσσ
+µρρρ, µρσσ
+ησxx
+µρxx
+ηρ = 2(c1,1 + c1,2)2 + 12¯φ2
+1 + 2χ′ ¯φ2
+2
+µσ = 2(c2,1 + c2,2)2 + 12¯φ2
+2 + 2χ′ ¯φ2
+1
+ησ = χ + α + 4χ′ ¯φ1 ¯φ2
+µρ = χ − α + 4χ′ ¯φ1 ¯φ2
+ηρρ = −6(c1,1 + c1,2) + 12¯φ1
+µσσ = −6(c2,1 + c2,2) + 12¯φ2
+ησσ = 2χ′ ¯φ1
+ηρσ = 4χ′ ¯φ2
+µρρ = 2χ′ ¯φ2
+µρσ = 4χ′ ¯φ1
+ηρσσ = µρρσ = 2χ′
+ηρρρ = µσσσ = 4
+ηρxx = µσxx = −κ
+Zimmermann [57]
+∂tA = −G∆
+�
+ε + (1 + ib)∆ − (1 + ic)|A|2�
+A
+with A = Ar + iAi
+ϱ = Ar
+ς = Ai
+ησ, µρ
+ηρρ, ηρσ, ησσ
+µρρ, µρσ, µσσ
+ηρ = µσ = −Gε
+ηρσσ = ηρρρ = µρρσ = µσσσ = G
+µρσσ = µρρρ = −ηρρσ = −ησσσ = −Gc
+ησxx = −µρxx = Gb
+ηρxx = µσxx = −G
+Reciprocal Cahn-Hilliard [variational structure]
+∂φi = ∆ δF
+δφi
+with F = �
+i Fi + Fcoup
+Fi = �
+i
+�
+dV
+�
+κi
+2 (∇φi)2 + αi
+2 φ2
+i + βi
+3 φ3
+i + γi
+4 φ4
+i
+�
+Fcoup =
+�
+dV
+�
+K∇φ1∇φ2 + aφ1φ2 + b1φ2
+1φ2 + b2φ1φ2
+2
++c1φ3
+1φ2 + c2φ2
+1φ2
+2 + c3φ1φ3
+2
+�
+1
+|V |
+�
+V dV φi = 0
+ϱ = φ1
+ς = φ2
+–
+ηρ = α1
+µσ = α2
+ηρρ = β1
+µσσ = β2
+ηρρρ = γ1
+µσσσ = γ2
+ηρxx = −κ1
+µσxx = −κ2
+ησ = µρ = a
+ησxx = µρxx = −K
+ηρσ = 2µρρ = 2b1
+2ησσ = µρσ = 2b2
+ηρρσ = 3µρρρ = 3c1
+ηρσσ = µρρσ = 2c2
+3ησσσ = µρσσ = 3c3
+TABLE III. Identification of models studied in the literature [16, 47, 56, 57] as special cases of the here derived general nonreciprocal Cahn-
+Hilliard model in the form of Eqs. (37). All literature models consider constant mobilities, i.e., Q = R = 1. Furthermore the relation to the
+reciprocal limit case is given.
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+16
+0
+km
+k +
+k
+0
+∆m
+∆, Ω
+∼ ε
+∼ ε4
+FIG. 3. Dispersion relations (43) below (blue lines), at (purple line) and above (red line) the threshold of a codimension-2 Cahn-Hilliard
+instability where two Cahn-Hilliard modes become simultaneously unstable. Labeled dotted lines and solid bars indicate typical quantities and
+scalings above onset.
+3.
+Amplitude equation for codimension-2 Cahn-Hilliard instability
+Up to here we have considered the conserved-Hopf instability as one basic codimension-1 bifurcation and have derived the
+nonreciprocal Cahn-Hilliard equations as the corresponding generic amplitude equation. Next, we furthermore show that it also
+corresponds to the amplitude equation of another (codimension-2) instability that involves two conservation laws. In particular,
+we consider the dispersion relation characterized by two real eigenvalues given by
+λ±(k) = δ±k2 − ˜δ±k4 + O(k6) .
+(43)
+Considering the codimension-2 instability where both real modes simultaneously become unstable at control parameter ε = 0,
+both leading order growth rates are small, i.e., δ± = δ±,2ε2. Fig. 3 illustrates the dispersion relations below, at and above the
+onset. The indicated scalings of the band of unstable wavenumbers and of the maximal growth rate are as in Fig. 1 of the main
+text.
+Again we consider the general multi-component model (11), naively use the same ansatz (12) and only discuss the differences
+for the current case as compared to the analysis done in Section 2 of the Supplementary Material. First, all terms at order ε and
+ε2 are unchanged. We also find the same linear system (20) at order ε3. However, in contrast to the previously treated case of
+the conserved-Hopf instability, the resulting eigenvalues are real, i.e., the inequality (24) is reversed and we identify
+Q0ηρ + R0µσ
+2
+±
+�
+(Q0ηρ − R0µσ)2
+4
++ Q0ησR0µρ = δ± .
+(44)
+Now we demand that both δ+ and δ− are O(ε2) and, thus, we can set them to zero at this order. In consequence, there is no
+dynamics on timescale τ which is as expected since there is no oscillation in contrast to the conserved-Hopf case. In other
+words, while for the case of a conserved-Hopf instability the leading order frequency is ωk2 with ω = O(1), here ω is O(ε2)
+and imaginary, i.e., is simply part of the growth rate.
+Proceeding to orders ε4 and ε5, equations (31)-(34) are recovered and using the same recombination as in Eq. (35), again
+we obtain amplitude equations that correspond to the generalized nonreciprocal Cahn-Hilliard model (37). Linearization yields
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+
+17
+Eq. (39), and calculating the eigenvalues, again we recover the original dispersion relation, i.e., we identify
+Q0ηρxx + R0µσxx
+2
+± ηρηρxxQ2
+0 + (2ησxxµρ + 2ησµρxx − ηρxxµσ − ηρµσxx)Q0R0 + µσµσxxR2
+0
+2
+�
+η2ρQ2
+0 − 2ηρµσQ0R0 + R0(4ησµρQ0 + µ2σR0)
+= −˜δ± .
+(45)
+We conclude that beside the conserved-Hopf instability the generalized nonreciprocal Cahn-Hilliard model also describes the
+generic behavior close to the simultaneous onset of two large-scale stationary instabilities with conservation laws, i.e., two Cahn-
+Hilliard instabilities. It is valid when the inequality (24) is reversed, i.e., when the imaginary part of the eigenvalues vanish, and
+the resulting additional contributions to the growth rates are still small, i.e., O(ε2).
+Note that, in general, there are no further emerging conditions on the coefficients of the nonlinear terms in Eq. (37), i.e.,
+despite the stationary character of the considered codimension-2 instability, the resulting amplitude equations are normally still
+nonreciprocal, i.e., nonvariational, if the original model is nonvariational. In practice, this implies that nonlinear states resulting
+from secondary, tertiary, etc. instabilities may exhibit time-dependent (periodic or irregular) behavior. In contrast, if the original
+model is variational itself the resulting amplitude equations will directly inherit the variational structure and the parameters of the
+nonreciprocal Cahn-Hilliard model aquire mutual relations that renders it a reciprocal Cahn-Hilliard model. The corresponding
+conditions for the various parameters are given in the final row of Table III. This occurs, for instance, in the case of dewetting
+isothermal two-layer liquid films on solid substrates [24, 42] and decomposing ternary mixtures [31, 32].
+Preprint– contact: u.thiele@uni-muenster.de – www.uwethiele.de – January 16, 2023
+

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+Random bit generation based on self-chaotic microlasers 
+with enhanced chaotic bandwidth 
+JIAN-CHENG LI,1,2JIN-LONG XIAO,1,2 YUE-DE YANG,1,2 AND YONG-ZHEN HUANG1,2, * 
+1State Key Laboratory of Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, 
+China 
+2Center of Material Science and Optoelectronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China 
+*Corresponding author Email: yzhuang@semi.ac.cn  
+Abstract: Chaotic semiconductor lasers have been widely investigated for high-speed random bit generation, which 
+is applied for the generation of cryptographic keys for classical and quantum cryptography systems. Here, we propose 
+and demonstrate a self-chaotic microlaser with enhanced chaotic bandwidth for high-speed random bit generation. By 
+designing tri-mode interaction in a deformed square microcavity laser, we realize a self-chaotic laser caused by two-
+mode internal interaction, and achieve an enhanced chaotic standard bandwidth due to the photon-photon resonance 
+effect by introducing the third mode. Moreover, 500 Gb/s random bit generation is realized and the randomness is 
+verified by the NIST SP 800-22 statistics test. Our demonstration promises the applications of microlasers in secure 
+communication, chaos radar, and optical reservoir computing, and also provides a platform for the investigations of 
+multimode nonlinear laser dynamics. 
+ 
+1. Introduction 
+Physical random bits play an important role in cryptography systems, information security, stochastic modeling, and 
+Monte Carlo simulation [1–4]. Physical random bit generation (RBG) was achieved with low generation rates (at Mb/s 
+level) based on thermal noise [5] and sampling phase jitter [6] in specific circuits, stochastic threshold switching in 
+memristors [7], and quantum vacuum state fluctuations [8]. To realize high-speed RBG, chaotic semiconductor lasers 
+as favorable physical entropy sources have been widely investigated owing to their large bandwidth and intensive 
+randomness [9–14]. However, semiconductor lasers, governed by the two parameters of mode intensity and carrier 
+inversion, usually need external perturbations to generate specific nonlinear dynamic states, such as periodic 
+oscillations and chaos [15]. Consequently, chaotic semiconductor lasers were investigated under external optical 
+feedback [9, 10, 16–18] and optical injection [14, 19, 20]. To simplify the system complexity, integrated chaos lasers 
+were developed with a passive feedback cavity [12, 21–23], optoelectronic feedback [24], or mutual injection lasers 
+[25, 26]. In addition, deterministic polarization chaos, caused by nonlinear mode competition including carrier spin 
+relaxation, was realized for a free-running quantum dot vertical-cavity surface-emitting laser [27]. Recently, parallel 
+ultrafast RBG was demonstrated in a broad area semiconductor laser with curved facets, using spatiotemporal 
+interference of many lasing modes with unpredictable spontaneous noise [28]. A self-chaotic microcavity laser was 
+demonstrated by using two-mode internal interaction, and 10 Gb/s RBG was obtained from the chaotic laser output 
+[29]. However, chaotic semiconductor lasers under delayed optical feedback or mutual coupling have obvious 
+correlation peaks of the time delay signature [9, 13], which reduces the randomness and security in random number 
+generation. The ultrafast RBG relying on many modes requires a large broad area cavity under large-current pulse 
+operation [28], and the chaotic bandwidth and RBG rate were limited by the laser relaxation oscillation frequency 
+[29]. 
+In this paper, we propose and demonstrate a tri-mode self-chaotic microlaser with an enhanced chaotic bandwidth 
+by employing the photon-photon resonance effect [30]. By designing a deformed square microcavity with circular 
+sides, we can enhance the mode Q-factors and engineer the mode frequency interval [31]. Since passive mode Q-
+factors are larger than 104 for the fundamental (0th), first (1st) and second-order (2nd) transverse modes, they can all 
+approach the threshold condition for an AlGaInAs/InP deformed square microlaser, with a Q-factor determined by 
+absorption loss much lower than 104. As shown in Fig. 1(a), the self-chaotic microlaser is realized by the mode 
+interaction between the 0th and 1st order transverse modes, and the chaotic bandwidth is enhanced due to photon-
+photon resonance caused by mode beating with the 2nd order transverse mode. The mode intensity patterns of the 0th 
+and 1st transverse modes are shown in the insets of Fig. 1(c), their field distributions are in-phase and anti-phase in 
+half a region, respectively, which is clearer than those in the deformed hexagonal microcavity [29]. The enhancement 
+and cancellation of mode beating intensities result in strong differences of carrier consumption in the in-phase and 
+anti-phase regions, which transfer at the mode beating frequency. As the beating frequency approaches the relaxation 
+
+oscillation frequency, the mode beating intensity will cause strong carrier oscillation and the appearance of mode side 
+peaks similar to that under external modulation, which results in strong internal mode interaction and self-chaos [29]. 
+The further mode beating with the 2nd order transverse mode will induce additional high-frequency peaks in the 
+response curve as shown in Fig. 1(a), i.e., the chaotic bandwidth enhanced by the photon-photon resonance effect for 
+directly modulated lasers [30]. Based on the novel method, we demonstrate a tri-mode self-chaos deformed square 
+microcavity laser with 33.9 GHz chaos bandwidth, and realize 500 Gb/s RBG from the chaotic microlaser output. 
+ 
+ 
+ 
+Fig. 1. (a) Schematic diagram of self-chaos due to two-mode interaction and chaotic bandwidth enhanced by photon-photon resonance of the mode 
+beating with the third mode. Carrier oscillation at the beating frequency of the 0th and 1st transverse modes 0 and 1 causes side peaks for the lasing 
+modes, which work as the internal optical injection terms and cause self-chaos. The chaotic bandwidth is extended to the high-frequency region 
+due to the photon-photon resonance with the 2nd transverse mode 2. (b) Three-dimensional schematic diagram and two-dimensional top-view of a 
+circular-sided square microcavity laser with a central hole and a ring electrode corresponding to a refractive index step Δn. (c) Mode Q-factor versus 
+mode wavelength. Insets are mode intensity distributions of the 0th, 1st and 2nd transverse modes. (d)Transverse mode intervals Δf01 and Δf12 and (e) 
+degenerated mode intervals Δf00 and Δf11 versus Δn. 
+2. Self-chaos generation 
+A three-dimensional schematic diagram and two-dimensional top-view of the deformed square microcavity laser are 
+shown in Fig. 1(b), where a central hole is applied to further control the transverse mode number. The transverse 
+electric (TE) mode characteristics are numerically investigated using a two-dimensional finite element method, for a 
+deformed square with the flat-side length a = 20 µm, circular-side deformation parameter δ = 2.17 µm, the width of 
+output waveguide d = 1.5 µm, the shift of the output waveguide h = 4√2 µm, and the radius of the central hole Rin = 
+5.5 µm. Two degenerated modes with nearly the same magnitude of Q-factors and mode field patterns are obtained 
+
+(a)
+V
+(b) BCB,layer
+Ohmic contact window10
+1547
+1548
+1549
+1550
+1551
+1552
+1553
+Ix1.
+041(d)30
+e)0.0F
+oo1o1o20J00
+10
+Z
+Af113(468101214
++
+△n (x0.001)
+An (x0.001)AA.interaction
+AlGalnAs
+InP substrate
+Quantum wellspf
+High-frequency oscillation
+Photon-photon
+resonanceChaos Bandwidth enhancement
+M
+106S1Ω-factol
+0103for each transverse mode. We give the results for the degenerated mode with a higher Q-factor in the following. As 
+shown in Fig. 1(c), the simulated mode Q-factors are 4.1 × 105, 6.9 × 104, and 1.7 × 104 for the 0th, 1st and 2nd order 
+transverse modes, respectively, with mode wavelengths of 1550.093, 1550.160, and 1550.265 nm. The corresponding 
+squared magnetic field distributions are shown in the insets of Fig. 1(c). In addition, a ring p-electrode with a width 
+of 4 µm is designed for fine adjustment of the mode frequency interval, with a refractive index step ∆n to simply 
+account for carrier and temperature distributions inside the resonator [32]. The calculated mode frequency intervals 
+Δf01 = f0th - f1st and Δf12 = f1st – f2nd and degenerate mode intervals Δf00 and Δf11 versus Δn are plotted in Figs. 1(d) and 
+1(e), respectively. The magnitude of Δf01 around 10 GHz is suitable for realizing a chaotic microlaser caused by 
+internal mode interaction [29]. In the range 0.003 < Δn < 0. 008, complex mode coupling results in a large splitting 
+for Δf11.  
+ 
+ 
+ 
+Fig. 2. (a) Schematic of the experimental setup for the test of nonlinear dynamic states. ISO, isolator; OSA, optical spectrum analyzer; EDFA, 
+erbium-doped fiber amplifier; OBPF, optical bandpass filter; PD, photodetector; ESA, electrical spectrum analyzer; OSC, real-time oscilloscope. 
+(b) Laser power and applied voltage versus injected current. Insets: SEM image of a deformed square microcavity and lasing spectra map with 
+respect to current. (c) Lasing spectra and (d) corresponding electric power spectra of steady, periodic, and chaotic states at 5.6, 6.6 and 8.8 mA, 
+respectively. 
+According to the designed microcavity parameters, circular-sided square microcavity lasers were realized using 
+an AlGaInAs/InP compressively-strained multiple quantum well laser wafer with the same manufacturing process as 
+in [29]. The microlasers were tested at a heat sink temperature of 289 K using the experimental setup shown in Fig. 
+2(a). The output power coupled into a tapered single-mode fiber (SMF) and the applied voltage versus continuous-
+wave injection current are plotted in Fig. 2(b), where the insets are the scanning electron microscope image of an 
+etched microcavity and the lasing spectra from 4 to 40 mA. A threshold current of 4 mA is estimated based on lasing 
+spectra. Nonlinear dynamics of the laser output were investigated, including lasing spectra, radio-frequency (RF) 
+spectra, and time domain signal. As shown in Fig. 2(c), three peaks at 1539.384, 1539.560 and 1539.876 nm are 
+observed at an injection current of 5.6 mA, with mode frequency intervals of 22 and 39.5 GHz. Comparing the 
+simulated results in Fig. 1(c), these peaks are identified as the 0th, 1st and 2nd transverse modes, respectively. The 
+corresponding RF spectrum is shown in Fig. 2(d), which almost coincident with noise floor at 5.6 mA. By increasing 
+the current to 6.6 mA, side peaks with an interval of ~0.04 nm (~5 GHz) are observed for the main lasing peaks, which 
+may be attributed to the mode beating between the degenerate modes of the 1st transverse mode as indicated by the 
+simulated results in Fig. 1(e). A sharp harmonic peak at 5 GHz appears in the corresponding RF spectrum in Fig. 2(d) 
+at 6.6 mA. At 8.8 mA, a broadened lasing spectrum appears due to strong mode interaction, similar as the chaotic 
+
+(c)
+(a)
+5.6 mA
+6.6 mA
+8.8 mA
+OSA
+OSC
+-40
+1 st
+(dBm)
+用
+Microcavity laser
+-50
+厦
+ISO
+ESA
+othInten
+M
+PD
+-70
+OBPF
+Bias-T
+00
+2nd
+-80
+1539.5
+1540.0 1539.5
+1540.0 1539.5
+1540.0
+2.5
+(b)
+Wavelength (nm)2.0
+noise floor
+20
+5.6 mA
+(Mn)
+6.6 mA
+80
+8.8 mA
+Intensity
+ower
+(dBm)
+1.0P
+-40
+Inten
+320
+Curr
+-60-
+0.5
+5
+10
+-100
+80
+1540
+1550
+1560
+Wavel
+ngth(nm)
+0.0
+10
+20
+30
+40
+50
+0
+0
+5
+10
+15
+Current (mA)
+Frequency (GHz)lasing spectrum in [29], which is mainly caused by the mode interaction between the 0th and 1st modes. The chaotic 
+standard bandwidth, which covers 80% of the total RF power [33], is calculated to be 9.6 GHz at 8.8 mA. 
+3. Chaotic bandwidth enhancement 
+The enhancement of chaotic bandwidth due to photon-photon resonance is demonstrated in Fig. 3. Here, the main 
+lasing modes jump to around 1550.5 nm with even high injection currents due to the current heating effect. As shown 
+in Figs. 3(a) and 3(b), a long-wavelength mode assigned as the 2nd mode increases much faster than other lasing peaks 
+with the current and becomes the main lasing mode at 20 mA, and the high frequency peaks at around 21 and 32 GHz 
+of the RF spectra are greatly enhanced. The calculated chaos standard bandwidths are 13.7, 28.2, and 33.9 GHz at 16, 
+18, and 20 mA, respectively. To clearly verify the effect of photon-photon resonance, we measured RF spectra for 
+filtered optical spectra as shown in Figs. 3(c) and 3(d). The RF spectra have small chaos standard bandwidths of 13.2 
+GHz and 8.1 GHz for the filtered optical spectra with the 0th plus 1st modes (0th + 1st) and 2nd mode, respectively. In 
+Fig. 3(c), the intervals between the 2nd mode peak and three evident peaks of the wide chaotic spectra are 0.184, 0.272, 
+and 0.320 nm, which contribute to three beating peaks at 22.8, 33.3, and 39.5 GHz for the RF spectrum in Fig. 3(d). 
+These results imply the origin of bandwidth enhancement due to mode beating with the 2nd mode. The AC waveform 
+of the chaotic laser output at 20 mA is plotted in Fig. 3(e), and the calculated autocorrelation function (ACF) is shown 
+in Fig. 3(f), with a half width at half maximum of 0.011 ns. The ACF has some minor peaks within 0.5 ns, but without 
+the time-delayed correlation peak observed in optical feedback chaotic lasers [34]. The modified Grassberger-
+Procaccia (G-P) algorithm is applied to quantify the complexity of the chaos signal [35, 36], and a correlation 
+dimension of 11.6 is obtained, which is nearly triple that in [29].  
+ 
+ 
+ 
+Fig. 3. (a) Lasing spectra and (b) corresponding RF spectra at 16, 18 and 20 mA. (c) Filtered lasing spectra, the arrows show different peak intervals, 
+and (d) corresponding RF spectra at 20 mA. (e) Irregular temporal waveform and (f) corresponding autocorrelation function for the chaotic output 
+at 20 mA. The inset in (f) represents the entire ACF curve for 1 μs.  
+4. Random bit generation 
+Furthermore, the self-chaotic microlaser was utilized to generate physical random numbers. The AC waveform signals 
+at 20 mA are collected with a 100 GSa/s sampling rate, and the intensity histogram distribution of the 500 µs long raw 
+data stream is illustrated in Fig. 4(a). The intensity distribution is asymmetric with an initial skewness of 0.40, which 
+is a typical feature of a chaotic semiconductor laser. The asymmetric distribution can result in bias in the generated 
+random sequence, and we adopted extra post-processing methods, including delay-subtracting and least significant 
+bits (LSBs) extraction, for RBG [10, 28]. Specifically, we subtract the original signal from its delayed signal to attain 
+a symmetric distribution. Considering the very low correlation coefficient at 0.5 ns in Fig. 3(f), we select a delay time 
+of 0.5 ns and plot the histogram distribution of the differential data in Fig. 4(b). The symmetry of the differential signal 
+
+20
+-30
+01QA1.701550.5
+1550.0
+1550.5
+1551.0
+412
+413
+414
+1550.0
+415
+416Navelength (nm)
+Wavelength (nm)
+Time (ns)-60
+-60
+1.0.- noise floor
+(b)
+noise rioor16 mA
+ond
+0.86 mA
+a18 mA
+-70
+-70
+0.6A
+Oth+1 st+2nde200
+ 400
+600
+800
+10008 mA10
+AO7
+oc
+20
+30
+0
+0.0
+0.4
+0.6
+0.8
+L.0Freouer
+Time (ns)20mA沁北is significantly improved with a skewness coefficient of 0.02. Then, the differential intensity is digitalized into 8-bit 
+binary numbers, and the LSBs method is adopted to destroy the residual correlations of adjacent bits and improve the 
+uniformity of the bit distributions. By retaining five LSBs, we can obtain a nearly uniform probability distribution, as 
+shown in Fig. 4(c). At the same time, the autocorrelation coefficient of the bit stream is less than 10-3 and remains at 
+the background level for any bit stream length in Fig. 4(d), indicating the removal of correlation between successive 
+bits. 
+ 
+ 
+ 
+Fig. 4. Histogram distribution for (a) raw signal intensity and (b) differential intensity after delay-subtracting post-processing. (c) Probability 
+distribution with 5-LSBs extraction, and (d) corresponding ACF curve of the bit stream. 
+ 
+Finally, the randomness of the generated random bits is verified using the NIST Special Publication 800-22 
+statistical tests, by dividing 1-Gbit data into 1000 sequences of 1-Mbit [37]. When the significance level is set to 0.01, 
+the randomness test is successful if the P-value is larger than 0.0001 and the proportion is within 0.99 ± 0.0094392. 
+For the test items that produce multiple P-values and proportions, the worst case is selected and shown in Table 1, and 
+the generated random bits successfully pass the 15 NIST sub-tests. The obtained maximum electrical-delay self-
+difference RBG rate is 500 Gb/s (100 GSa/s×5 bits).  
+We also conducted an optical-delay self-difference experiment for random bit generation via balanced-detection 
+method in [38]. As shown in Fig. 5, the chaotic light from the microcavity laser at 20 mA is firstly amplified and 
+filtered. Then, the light is split into two paths after a 50:50 fiber coupler (FC). Delayed fiber (DL) with the length of 
+1 m (corresponding to 5 ns optical delay) is introduced into one of the two paths. The two beams are simultaneously 
+detected by a balanced detector (Finisar BPD V2120R, 43 GHz bandwidth). Then the converted electrical signal is 
+collected by the real-time oscilloscope at 100 GSa/s sampling rate. Finally, a least-significant-bits method is adopted 
+and 5-LSBs are selected to generate 500 Gb/s physical random number sequence. Similarly, 1000 sequences of 1-
+Mbit stream are set to the NIST SP 800-22 randomness test. All sub-tests are successful and shown in Table 1.  
+ 
+ 
+ 
+Fig. 5. Experimental setup for optical-delay self-difference random bit generator.  
+ 
+
+X103
+X10°
+12
+8robal
+ocorrelati
+0
+0.0100
+0010
+20
++0
+100
+Index
+Bit stream length NMicrocavitylaser
+OBPF
+EDFA
+ISO
+PD
+OSC
+FC
+10000
+DL
+PD
+Balanceddetector9
+6S
+S20
+0
+-40
+-20
+0
+20
+40
+-40
+-20
+0
+20
+40Voltage (mV)
+Voltage (mV)
+n 100CB
+-0-LSBs=5
+0.03u
+F
+000Table 1. NIST SP 800-22 test results for random bit sequence 
+Statistical Test 
+Electrical-delay self-
+difference RBG 
+Optical-delay self-
+difference RBG 
+  Result 
+P-value 
+Proportion 
+P-value 
+Proportion 
+Frequency 
+0.32214 
+0.985 
+0.60799 
+0.992 
+Success 
+Block Frequency 
+0.27027 
+0.989 
+0.80556 
+0.986 
+Success 
+Runs 
+0.37701 
+0.987 
+0.16080 
+0.989 
+Success 
+Longest Run 
+0.29109 
+0.989 
+0.14532 
+0.986 
+Success 
+Rank 
+0.62255 
+0.983 
+0.44655 
+0.992 
+Success 
+FFT 
+0.40296 
+0.987 
+0.14781 
+0.988 
+Success 
+Nonoverlaping template 
+0.00798 
+0.989 
+0.00487 
+0.986 
+Success 
+Overlapping template 
+0.30266 
+0.986 
+0.30412 
+0.990 
+Success 
+Universal 
+0.60177 
+0.987 
+0.11606 
+0.990 
+Success 
+Linear complexity 
+0.23927 
+0.989 
+0.14125 
+0.986 
+Success 
+Serial 
+0.13650 
+0.990 
+0.13572 
+0.991 
+Success 
+Approximate entropy 
+0.01395 
+0.993 
+0.99743 
+0.990 
+Success 
+Cumulative sums 
+0.04365 
+0.986 
+0.14206 
+0.987 
+Success 
+Random excursions 
+0.12120 
+0.986 
+0.01572 
+0.987 
+Success 
+Random excursions variant 
+0.05205 
+0.986 
+0.00714 
+0.995 
+Success 
+ 
+5. Conclusion 
+In summary, tri-transverse-mode lasing with mode intervals of around 10 to 30 GHz has been demonstrated in a 
+deformed square microcavity laser. The strong mode interaction between the 0th and 1st order transverse modes with 
+the oscillation of the local photon density distribution inside the microcavity results in self-chaotic laser output, as in 
+Ref. [29]. Moreover, the 2nd order transverse mode can induce additional beating peaks for the chaotic RF spectrum 
+and greatly enhance the chaotic signal bandwidth. Based on tri-mode chaotic output, we have realized 500 Gb/s 
+physical random number generation using electrical and optical delay-subtracting RBG schemes. Our work paves the 
+way for mode engineering to enhance the self-chaotic bandwidth for deformed microcavity lasers. A random number 
+generator based on the self-chaos deformed square laser can simplify the system greatly due to a small footprint and 
+low power consumption for the chaotic microlasers. Moreover, self-chaotic lasers have potential applications in secure 
+communication, chaos radar and optical reservoir computing. 
+ 
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+

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