diff --git "a/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/load_file.txt" "b/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/1dE3T4oBgHgl3EQfnQrW/content/tmp_files/load_file.txt" @@ -0,0 +1,2603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf,len=2602 +page_content='Typical Correlation Length of Sequentially Generated Tensor Network States Daniel Haag,1, 2, 3, ∗ Flavio Baccari,1, 3 and Georgios Styliaris1, 3 1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1, 85748 Garching, Germany 2Physik-Department, Technische Universität München, James-Franck-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1, 85748 Garching, Germany 3Munich Center for Quantum Science and Technology (MCQST), Schellingstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4, 80799 München, Germany (Dated: January 12, 2023) The complexity of quantum many-body systems is manifested in the vast diversity of their cor- relations, making it challenging to distinguish the generic from the atypical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This can be addressed by analyzing correlations through ensembles of random states, chosen so as to faithfully embody the relevant physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here we focus on spins with local interactions, whose corre- lations are extremely well captured by tensor network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Adopting an operational perspective, we define ensembles of random tensor network states in one and two spatial dimensions that admit a sequential generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As such, they directly correspond to outputs of quantum circuits with a sequential architecture and random gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In one spatial dimension, the ensemble explores the entire family of matrix product states, while in two spatial dimensions, it corresponds to random isometric tensor network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We extract the scaling behavior of the average correlations between two sub- systems as a function of their distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using elementary concentration results, we then deduce the typical case for measures of correlation such as the von Neumann mutual information and a measure arising from the Hilbert–Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We find for all considered cases that the typical behav- ior is an exponential decay (for both one and two spatial dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We observe the consistent emergence of a correlation length that only depends on the underlying spatial dimension and not the considered measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Remarkably, increasing the bond dimension leads to a higher correlation length in one spatial dimension but has the opposite effect in two spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' INTRODUCTION The behavior of correlations in quantum many-body systems is an inherently difficult problem to character- ize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Specifying a generic n-particle state requires ex- ponentially many parameters, a fact which reflects the enormous variety of correlations possible in the quantum realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Nonetheless, significant insights can be gained about the nature of correlations by utilizing random en- sembles of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A celebrated result along this direction shows that random states sampled uniformly from the full Hilbert space of an n-particle system typically ex- hibit strong correlations, as manifested by a volume law behavior for the entanglement entropy [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, there is by now clear evidence that the set of physically relevant states constitutes an exponentially small subset of the full Hilbert space of an n-particle system [6], bring- ing into question the relevance and utility of conclusions obtained under the assumption of uniform sampling from the full Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For quantum spin systems with local interactions, tensor network states have been exceedingly successful at capturing relevant properties [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They exhibit an area law for the entanglement entropy by construction, and are, therefore, good candidates to represent many physically relevant many-body states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Their preeminent one-dimensional representatives, matrix product states (MPS), have been shown to represent faithfully ground states of gapped local Hamiltonians [8–10] and have given ∗ daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='haag@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='de rise to the complete classification of topological phases of matter in one dimension [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' MPS have been also generalized to their counterparts in two (or more) spa- tial dimensions, projected entangled pair states (PEPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While only a weaker link between local Hamiltonians and PEPS has been proven rigorously, two-dimensional PEPS are known to efficiently represent a wide class of strongly correlated states [7, 13], including states with power-law [14] and topological correlations [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The importance of defining ensembles of random tensor network states for the purpose of exploring typical prop- erties of physically relevant states has been recognized more than a decade ago [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' MPS ensembles have been utilized to gain insights into, among other things, the typ- icality of expectation values of local observables [18, 19], equilibration under Hamiltonian time evolution [20], the entropy of subsystems [21], non-stabilizerness [22], and, most relevant for this work, the behavior of correla- tions [23–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, correlation functions of ran- dom inhomogeneous MPS (that is, MPS whose local tensors can be different) were shown to exhibit almost surely an exponential decay [24, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A qualitatively similar behavior was observed also for correlation func- tions of translation-invariant MPS and PEPS with ran- dom Gaussian entries [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Instead of incorporating the randomness directly at the level of states, one can also consider random local Hamiltonians and examine their ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The typical behavior of correlations for this case was found to depend on the nature of random- ness, allowing for both long- and short-range correlated states [23, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here, we approach the problem of typical correlations arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='04624v1 [quant-ph] 11 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Sequential generation of MPS and isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each circle represents a site of the finalized state and boxes represent the isometries of the sequential generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (1a) Sequential generation of an MPS with physical dimension d and bond dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The diamond indicates the origin of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (1b) Each isometry arises from a unitary matrix of U(dD), with input and output as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The (blue) ancillary system is initialized at the first step of the sequential generation, transferred along the process, and accumulated at the final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (2a) Sequential generation of an isoTNS with physical dimension d and bond dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition to indicating the origin of the process, the diamond also indicates the orthogonality center of the isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (2b) Each isometry arises from a unitary matrix of U � dD2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Ancillary systems are initialized and eventually accumulated at the boundary of the isoTNS at different steps of the sequential generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' in random MPS and PEPS from a more operational point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We introduce families of inhomogeneous ran- dom tensor network states that arise from a sequential generation in a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Such ensembles are, by definition, directly connected to the study of quan- tum circuits with a sequential architecture and random gates, where each unitary gate is independently cho- sen randomly according to the uniform (Haar) measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the one-dimensional case, every MPS admits such a preparation [31], where the bond dimension dictates the number of overlapping qudits between any two succes- sive gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the two-dimensional setting, our ensemble can be understood as being uniform over the space of so-called isometric tensor network states (isoTNS) [32], which are PEPS with given bond dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In this case, the resulting family of random circuits is composed of two-dimensional circuits with local overlapping gates, each resembling a tile acting on a neighborhood of qu- dits [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although isoTNS are a subfamily of PEPS, they are known to contain a rich variety of strongly cor- related states, such as topological models [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For the above ensembles, we study the scaling behavior of the average correlations between two subsystems as a function of their distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We then utilize this average behavior of correlations to deduce the typical case via concentration inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Instead of using well-known correlation functions, we perform the analysis using a measure of correlation arising from the Hilbert–Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although in a generic many-body setting such a measure might have undesirable properties, we show that it is particularly suited in the context of tensor network states because it bounds the trace distance as well as all connected correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For MPS, we also con- sider the Rényi-α mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given a technical conjecture, we compute the average correlations for all integer values of α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We then use those results to retrieve the von Neumann mutual information [35] via analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First, we confirm analytically the common intuition that inhomogeneous MPS typically exhibit exponentially decaying correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show that a single common correlation length ξ1D persists among different measures of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We obtain a similar quantitative conclu- sion for two-dimensional isoTNS, where we observe the emergence of a different correlation length ξ2D that is also consistent among different measures of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Both lengths have a rather weak dependence on the underly- ing bond dimension D of the tensor network and remain short-range correlated for all values of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Surprisingly, however, ξ1D and ξ2D have exactly opposite behaviors when D is varied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ξ1D monotonically increases while ξ2D monotonically decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our findings also establish that exponentially decaying correlations are typical for the family of (inhomogeneous) isoTNS, and consequently for the random states produced by the corresponding quan- tum circuit architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II, we in- troduce our families of sequentially generated tensor net- work states and the main technical tools required to com- pute their average properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' III, we summarize our results for both MPS and isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV and V, we respectively discuss our findings for MPS and isoTNS in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lastly, we devote Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' VI to final observations and potential follow-ups to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' PRELIMINARIES In this section, we introduce the main technical con- cepts that will be needed throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II A, we review the relevant families of sequentially generated tensor network states in one and two dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II B is devoted to the measures of correlation we are interested to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II C, we explain how to compute averages with respect to the Haar mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lastly, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II D introduces the graphical notation we will use to present and prove our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Tensor Network States In one dimension, the preeminent tensor network struc- ture are matrix product states (MPS) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' An n-particle MPS with open boundary conditions and local (physical) dimension d is given by |ψ⟩ = � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',in ⟨L|A(1) i1 · · · A(n) in |R⟩|i1 · · · in⟩, (1) where A(j) ∈ CD×D, |L⟩ ∈ CD is the left boundary con- dition, and |R⟩ ∈ CD is the right boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D is called the bond dimension of the MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A commonly used graphical notation for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (1) is |ψ⟩ = , (2) where vertical (red) legs represent physical space indices � Cd� , and (blue) legs represent bond space indices � CD� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From its definition, it might not be evident how an MPS can be prepared because each tensor A does not necessarily correspond to a physical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, the representation of an MPS in terms of tensors is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This can be resolved by imposing a convenient canonical form [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Any MPS in such a canonical form can be seen as a state generated sequentially by applying unitary matrices U (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , U (n) ∈ U(dD) to a product state initialized in |0⟩⊗n for the physical space and |0⟩ for the bond space [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The resulting state is given by |ψ⟩ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (3) Note that the final site has dimension dD, while all other sites have dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we will see later, the final site will not play a significant role in our analysis, making its different dimension not an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (1a), we sketch an equivalent representation of sequential generation, in terms of isometries instead of unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The family of MPS is thus equivalent to states re- sulting from quantum circuits that have a sequential architecture and act on input product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The ar- chitecture is a consequence of the connectivity of the MPS network [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (1a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In this picture, larger bond dimensions translate to wider gates, each acting on 1+⌈logd(D)⌉qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For example, for D = d2 and n = 4, one has |ψ⟩ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (4) Naturally, using this correspondence, all of our results can be translated to the language of quantum circuits with the described architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Projected entangled-pair states (PEPS) are the gener- alization of MPS to two (or more) dimensions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Be- cause no simple generalization of the sequential gener- ation of MPS to arbitrary PEPS is known, we restrict ourselves to the rich family of two-dimensional isometric tensor network states (isoTNS), which were first defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [32] (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Much like MPS, isoTNS can be generated sequentially by applying unitary matrices to a product state initial- ized in |0⟩⊗mn for the physical space [33], where m de- notes the number of rows and n the number of columns of the underlying rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will use the sequential generation sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a), which is a generalization of the one proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each box corresponds to an isometry that arises from a unitary matrix U (i,j) ∈ U � dD2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, isometries in the bulk can be drawn as , (5) as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The diamond in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a) in- dicates the so-called orthogonality center of the isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Its row and column constitute the orthogonality hyper- surface, which can be treated like an MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, if an operator is supported only on the orthogonality hyper- surface, its expectation value with respect to the isoTNS reduces to that of the underlying MPS [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although isoTNS of a given bond dimension form by definition only a subclass of PEPS, they are known to contain states with a rich structure of correlations, such as topological mod- els [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On top, their properties make isoTNS a suitable candidate for studying correlations analytically, which is otherwise a generally challenging task in more than one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IsoTNS correspond to quantum circuits on a two- dimensional grid with local overlapping gates, which now resemble tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Increasing bond dimension translates to larger tile sizes and overlaps, as in the MPS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The corresponding architecture is dictated by the connectiv- 4 ity of the isoTNS network [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a)], and it is te- dious (although straightforward), which is why we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [33] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Quantifying Correlations Correlations express that knowledge about one sub- system can convey information about another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They are quantified by different measures that frequently arise from an information-theoretic perspective and are based on operationally motivated tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A prime example is the von Neumann mutual information [35] I(A : B) = S(ϱA) + S(ϱB) − S(ϱAB), (6) where S(ϱ) = − tr[ϱ log(ϱ)] (7) is the von Neumann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A and B are two disjoint subsystems of a larger system, and ϱA and ϱB denote the marginals of ϱAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The von Neumann mutual information captures the total (classical and quantum) amount of cor- relations between A and B, as it is equal to the minimum rate of randomness required to asymptotically turn ϱAB into a product state [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is also a non-negative quan- tity and non-increasing under local operations [35], both desirable properties for a measure of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The latter means that a quantum channel [35] acting on A or B alone (for example, by discarding part of a subsystem) cannot increase I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Unfortunately, the analytical treatment of the von Neumann mutual information is im- practical because computing the logarithm of ϱ generally requires the knowledge of its full spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To overcome this issue, an alternative is to consider a particular Rényi-α generalization of the mutual informa- tion Iα(A : B) = Sα(ϱA) + Sα(ϱB) − Sα(ϱAB), (8) where Sα(ϱ) = 1 1 − α log[tr(ϱα)] (9) is the Rényi-α entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As is apparent from the defini- tion, for integer values of α, its evaluation is considerably simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The Rényi-α mutual information has been inves- tigated in the context of conformal field theories [39–41], free fermions [42], and quantum dynamics [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will use later that the α → 1 limit of Iα(A : B) recovers the von Neumann mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The mentioned positive aspects notwithstanding, unlike the von Neu- mann mutual information, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (8) does not arise from a (generalized) divergence [45], and the Rényi-α mutual information can be negative [46, 47] and increasing under local operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is thus hard to interpret it as a proper measure of correlation in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Nevertheless, for cer- tain families of initial states (see, for example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [44]) monotonicity and non-negativity can be restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Hence- forth, we will mostly focus on the case of α = 2, but we will also consider an analytic continuation on positive in- teger values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we will show, in the present context of tensor network states, the case of α = 2 appropriately captures the decay of correlations at large distances be- tween subsystems A and B with little effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition to the previous quantities, we would also like to probe the trace distance T(A : B) = 1 2∥ϱAB − ϱA ⊗ ϱB∥1, (10) where ∥ · ∥p denotes the Schatten p-norm [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For an operator X, the Schatten p-norm is given by ∥X∥p = tr �� X†X �p/2�1/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (11) T(A : B) has a well-known operational interpretation, as it quantifies the optimal distinguishability between ϱAB and the product of its marginals ϱA⊗ϱB by a two-element generalized (global) measurement [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moreover, the trace distance upper bounds the (properly normalized) connected correlation function [45]: T(A : B) ≥ 2|⟨MA ⊗ MB⟩ − ⟨MA⟩⟨MB⟩| ∥MA∥∞∥MB∥∞ (12) Although the bound can be tight, the two quantities are different whenever product measurements are ineffective in distinguishing ρAB from ρA ⊗ ρB, a fact used in quan- tum data hiding [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As one expects from its operational interpretation, the trace distance satisfies the monotonicity property under local operations [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, T(A : B) is usually hard to compute exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will now argue that investigating N(A : B) = ∥ϱAB − ϱA ⊗ ϱB∥2 2 (13) meaningfully probes T(A : B) for tensor network states with constant bond dimension, all while being much sim- pler to treat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In general, for mixed many-body states, the two mea- sures can have vast disagreement because it holds [48] that ∥X∥2 ≤ ∥X∥1 ≤ � rank(X)∥X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (14) Both bounds are tight, and the upper bound is saturated for X ∝ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As such, for arbitrary mixed states of an exponentially large Hilbert space, the factor rank(X) can render the upper bound useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Crucially, in this work we investigate (random) tensor network states with fixed bond dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let ∂R denote the boundary of a system R and |∂R| its size (number of sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The ranks of ϱA and ϱB are respectively upper bounded by D|∂A| and D|∂B|, and that of ϱAB by D|∂A|+|∂B| [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, rank(ϱAB − ϱA ⊗ ϱB) ≤ 2D|∂A|+|∂B|, (15) 5 yielding the bound 1 2 � N(A : B) ≤ T(A : B) ≤ � D|∂A|+|∂B| 2 � N(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (16) For MPS (one dimension) with connected subsystems A and B, the bound reads 1 2 � N(A : B) ≤ T(A : B) ≤ D2 √ 2 � N(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (17) That is, the bound is independent of the sizes of A and B, unlike in the generic case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This suggests that, for reasonably small bond dimension, N(A : B) is a reliable probe of correlations [as quantified by T(A : B)] between subsystems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will expand on this point later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' k-Fold Twirl Let X be an operator acting on (Cq)⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The k-fold twirl of X with respect to the Haar measure on the uni- tary group U(q) is defined [51–53] as T (k) U (X) = � dU U ⊗kX � U †�⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (18) One can employ the Schur–Weyl duality for unitary groups to show [51, 54] that T (k) U (X) = � σ,τ∈Sk Wg � στ −1, q � P (q) σ tr � X � P (q) τ �T � , (19) where P (q) π : v1 ⊗ · · · ⊗ vk �→ vσ−1(1) ⊗ · · · ⊗ vσ−1(k) (20) is the representation of π ∈ Sk on (Cq)⊗k, where Sk is the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Wg � στ −1, q � = � G−1� στ [55] is the Weingarten function, where G ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' denotes the Gram matrix whose entries are given by Gστ = tr � P (q) σ � P (q) τ �T � = q#(στ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (21) Above, #(π) counts the number of cycles in the decom- position of π ∈ Sk into disjoint cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, Wg(π, q) depends only on the conjugacy class of π [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A, we show how to obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (19) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (18) by using a result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Graphical Notation In this section, we introduce the graphical notation used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To keep the images compact, we employ the operator-vector correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let {|i⟩} denote the standard basis of Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, the operator- vector correspondence [48] is defined by vec(|i⟩⟨j|) = |i⟩ ⊗ |j⟩ (22) and extended linearly to the vector space at large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we consider the standard (product) basis to be fixed, we do not distinguish between tensors (as mul- tidimensional arrays) and their basis-independent coun- terparts (such as vectors and operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let X be an operator acting on (Cq)⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the operator-vector correspondence, we denote it by = vec(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (23) Note that the orientation of the legs does not have any meaning in our images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, = = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (24) When we need the transpose of an operator, we will ex- plicitly use = vec � XT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (25) As such, when we contract two operators X and Y , we mean the trace of their product: = tr(XY ) (26) Let us state the two most prominent operators we will come across.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will see = vec � P (q) σ � , (27) where the horizontal (green) leg is permutation valued, and = vec � |0⟩⟨0|⊗k� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (28) Their relevant contractions are = tr � |0⟩⟨0|⊗kP (q) σ � = 1 (29) and = tr � P (q) σ P (q) τ � = q#(στ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (30) Moving forward, we will not explicitly write the oper- ator vec as it shall be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We investigate average correlations between two sub- systems A and B as a function of their (horizontal) distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A and B respectively stretch across a and b consecutive (horizontal) sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (a) The diamond indicates the origin of the sequential generation of the MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (b) In addition to indicat- ing the origin of the sequential generation, the diamond also indicates the orthogonality center of the isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For now, we restrict ourselves to A and B that touch the orthogonality hypersurface and stretch across h consecutive vertical sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With the definition of the Weingarten matrix, = Wg � στ −1, q � , (31) we can then write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (19) as T (k) U (X) = , (32) where the contraction of two green legs corresponds to a summation over the permutations of Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' SUMMARY OF RESULTS In this paper, we analyze the average behavior of cor- relations in random tensor network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Through the average, we obtain conclusions about the generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our work focuses on the disordered case, that is, the case where each local tensor is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our setting can be equivalently understood as an investigation of corre- lations in states resulting from quantum circuits with a sequential architecture and random gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In one dimension, generic MPS are known to exhibit exponentially decaying correlations in the translation- invariant case [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This is due to the fact that injectivity is a generic property [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On the other hand, injectivity alone is not enough to guarantee exponential decay of cor- relation for an inhomogeneous sequence of tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Nev- ertheless, the exponential decay of correlations is widely expected to persist without translational invariance but has never been rigorously studied so far in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In two (or more) dimensions, the landscape of correla- tions is much richer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For instance, already in two dimen- sions, certain PEPS corresponding to thermal states of classical models are known to exhibit power-law correla- tions [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moreover, prominent topological states, such as quantum double models [57] (which include the toric code) and string-net models [16, 17], admit a description in terms of PEPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On the other hand, for translation- invariant PEPS whose tensors’ entries are drawn from a Gaussian measure, it is known that correlations typically decay exponentially [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Computing correlations in higher-dimensional systems usually poses a significant challenge because they can be mediated through different paths connecting the two sub- systems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here, we restrict our analysis to two-dimensional isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This rich class of tensor net- work states is relevant in both the analytical and the numerical context [58–62], all while admitting a simple physical interpretation through sequential generation [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moreover, its mathematical properties make the analytical study of correlations in two dimensions tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For isoTNS, it is expected that correlations between two subsystems decay exponentially if they are both on the orthogonality hypersurface because the calculation reduces to the contraction of an MPS [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Nonetheless, isoTNS can represent a rich variety of topological mod- els, as all string-net models admit an exact and explicit description in terms of isoTNS [34] (on the appropriate underlying lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This motivates us to study the typ- ical behavior of correlations in isoTNS, particularly be- tween subsystems that extend beyond the orthogonality hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To investigate the decay of correlations in our two fam- ilies of random tensor network states, one must specify the ensembles to draw from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here we adopt an oper- ational perspective and relate our measures of random- ness directly to the sequential generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Be- cause that is defined with respect to isometries, one can incorporate randomness at the level of the under- lying unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A natural choice is to draw each unitary from the Haar measure on the appropriate uni- tary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This approach was introduced for MPS in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [18] (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [19]), and it can directly be applied to higher-dimensional tensor network states that admit a sequential generation, such as isoTNS, yielding normal- ized states by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although one can sample random translation-invariant states with this method, we investigate the disordered case by drawing each unitary matrix independently from the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we are interested in the decay of correlations, we focus on computing average correlations between two subsystems A and B as a function of their distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For 7 random MPS and isoTNS, we consider subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In both cases, A and B stretch across a and b consecu- tive (horizontal) sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (b), A and B touch the orthogonality hypersurface and stretch across h vertical sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will relax this condition later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For all of the measures of correlation we study, we find that the average with respect to the considered ensemble of states decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We formalize this type of behavior in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let C(A : B) denote a measure of correla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We say that the average of C(A : B) with respect to a given ensemble of random states decays exponentially if EC(A : B) = K exp � −r ξ � + O � exp � − r χ �� , (33) where K is constant with respect to r, and ξ > χ is the average correlation length for C(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Remarkably, we find that a single average correlation length persists throughout the different families of mea- sures of correlation and that it depends only on the un- derlying spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We later pinpoint the origin of this behavior to the invariance of a spectral gap of the corresponding family of transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For MPS, ξ = − � log � dD2 − d d2D2 − 1 ��−1 ≡ ξ1D, (34) and for isoTNS, ξ = − � log �dD3 − dD d2D4 − 1 ��−1 ≡ ξ2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35) Note that the average correlation length for MPS co- incides with that for isoTNS for d → dD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we will see later, this seemingly small modification changes the qualitative behavior substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Before moving to the detailed presentation of our methods and results, we briefly comment on the consid- ered measures of correlation and the implications of our findings, first for MPS and then for isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Results in One Dimension (MPS) In one dimension, we compute the averages of the Rényi-2 mutual information I2(A : B), the 2-norm ex- pression N(A : B), and the von Neumann mutual infor- mation I(A : B) (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II B for the definitions of the measures of correlation), with subsystems A and B as sketched in Fig 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We find that the averages decay exponentially as spec- ified in Definition 1 with the same correlation length ξ1D (see Results 1, 2, and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The derivation for I(A : B) relies on a technical conjecture (see Conjecture 1), which we will discuss in detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition, we show that the same conjecture is enough to assert that ξ1D is also the average correlation length for Iα(A : B) for any inte- ger value of α ≥ 1 (see Corollary 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In short, the same average correlation length ξ1D per- sists across different measures of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Interest- ingly, ξ1D depends very weakly on the bond dimension because, for d, D ≥ 2, ξ1D = � log � d ζ1D(d, D) ��−1 (36) with 3 4 ≤ ζ1D(d, D) < 1 (37) is monotonically increasing with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, it holds that limD→∞ ξ1D = 1/ log(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we are concerned with random tensor network states, ξ1D is obtained after averaging over realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is then natural to ask if exponentially decaying corre- lations are typical and, if so, what is the typical correla- tion length for an individual realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This motivates the investigation of the concentration of the distribution around its average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, we will show that it is exponentially unlikely in r that N(A : B) and I(A : B) decay slower than with ξ1D (see Corollaries 1 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our result for N(A : B) allows us to deduce that the average of the trace distance T(A : B) decays at least exponen- tially with correlation length ξ ≤ 2ξ1D, and it leads to a concentration result for T(A : B) (see Corollary 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Results in Two Dimensions (isoTNS) In two dimensions, we compute the averages of the Rényi-2 mutual information I2(A : B) and the 2-norm expression N(A : B), where subsystems A and B as sketched in Fig 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we find that the averages decay exponentially as specified in Definition 1 with the same average correlation length ξ2D (see Results 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The correlation length ξ2D displays a qualitatively dif- ferent dependence on the bond dimension that is albeit also rather weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, for d, D ≥ 2, ξ2D = � log � d ζ2D(d, D) ��−1 (38) with 0 < ζ2D(d, D) ≤ 8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (39) In contrast to its one-dimensional counterpart, ξ2D is a monotonically decreasing function of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As such, perhaps somewhat surprisingly, the largest correlation length is achieved for D = 2, which is still rapidly de- caying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For N(A : B), we can extend the applicability of our results to any size and shape of subsystems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 8 The decay is at least exponential with correlation length ξ = ξ2D (see Corollary 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We furthermore prove a con- centration result for N(A : B) expressing that it is highly unlikely that N(A : B) decays slower than with ξ2D (see Corollary 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This also allows us to draw a similar con- clusions about the behavior of T(A : B) (see Corollary 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' CORRELATIONS IN ONE DIMENSION In this section, we state and discuss the results for random MPS summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' III A in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Before doing that, we develop the tools behind our proofs in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A and IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV C, we compute the average of I2(A : B), and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV D, we investigate the decays of N(A : B) and T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we discuss the behavior of I(A : B) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' When computing the averages of measures of corre- lation for random MPS, we will exploit a simplification with respect to the scenario depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In- stead of allowing for an arbitrary number of sites be- fore subsystem A, we prove our statements in the limit c → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' J, this does not constitute a limitation because the c initial sites do not affect the decay of correlations and, therefore, neither the average correlation length ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Furthermore, we will see that the f sites after subsystem B do not play a role in the com- putation of average correlations, as it is expected for any sequentially generated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Transfer Matrices The key challenge for computing the average of each measure of correlation will be evaluating multiple expres- sions of the form tr � PE|ψ⟩⟨ψ|⊗k� , (40) where P = � P (d) e �⊗c ⊗ � P (d) α �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) β �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (41) The permutation α ∈ Sk acts on the sites comprising subsystem A, while β ∈ Sk acts on the sites comprising B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The exact forms of α and β as well as the number of required replicas k depends on the considered measure of correlation and will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It shall also be- come clear why sites belonging to neither A nor B are acted upon by the trivial permutation e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the fol- lowing, we show that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40) for random MPS reduces to multiplying matrices Tρ ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' with ρ ∈ Sk whose definition will be all but natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because their role is analogous to the known transfer matrices mediating cor- relations, we will also adopt this term here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Before introducing the transfer matrices, we must an- alyze E|ψ⟩⟨ψ|⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, let us define V (j) = U (j) �� U (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (3), |ψ⟩⟨ψ| = (42) and |ψ⟩⟨ψ|⊗k = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (43) By computing the k-fold twirl [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (32)], we obtain the building block = � dU (44) = , (45) where the (green) dot represents a Kronecker delta on three permutation indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Note that we have not drawn the contraction of a permutation matrix with |0⟩⟨0|⊗k because it is trivial by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of a random MPS is then given by E|ψ⟩⟨ψ|⊗k = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (46) We could, in principle, work with the building block above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, it is not convenient to have dangling bond (blue) legs whose dimension grows with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By cutting permutation-valued (green) legs instead, we ob- tain a building block with fixed dimension for fixed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With that building block, the average of a random MPS is given by E|ψ⟩⟨ψ|⊗k = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (47) The entries of the initial vector ⟨Ik| ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' are given by = = 1, (48) 9 where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The tensors in the bulk are given via = , (49) and the final tensor is given via = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (50) Computing an expression of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40) cor- responds to contracting each S with some P (d) ρ , which leads us to the promised definition of a transfer matrix Tρ ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='. Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (30), its entries are given by = = (51) = � σ∈Sk Wg � στ −1, dD � d#(σρ)D#(σθ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (52) We define Tρ with respect to the basis defined by the map si �→ ei, where si is the ith element of Sk = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , sk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' }, and {ei} is the standard basis of Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='. In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D, we find that Tρ is block triangular if the elements of Sk are ordered in a certain way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As alluded to in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (41), the final tensor S′ will be contracted with the trivial permutation e ∈ Sk in our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The final vector |Fk⟩ ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' is thus defined via = = (53) = � σ∈Sk Wg � στ −1, dD � (dD)#(σe) = δeτ , (54) where we have used the definition of the Weingarten func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the definitions of Te and |Fk⟩, it is easy to con- firm that Te|Fk⟩ = |Fk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Graphically, this implies the simplification = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (55) From this, it also follows that E|ψ⟩⟨ψ|⊗k is properly nor- malized: tr � E|ψ⟩⟨ψ|⊗k� = ⟨Ik|T n−1 e |Fk⟩ = ⟨Ik|Fk⟩ = 1 (56) With what we have laid out above, we can write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40) in terms of transfer matrices: tr � PE|ψ⟩⟨ψ|⊗k� = ⟨Ik|T c e T a αT r e T b β|Fk⟩ (57) We provide a simple Mathematica package [63] that defines Tρ with ρ ∈ Sk for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , 20} according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The package relies on the package provided by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [64] for evaluating the Weingarten function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Estimating the Decay of Correlations The decay of the average of each measure of correla- tion is necessarily determined by the r sites separating subsystems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we will see in the following sec- tions, this will, for each measure, translate to a simple statement in terms of the just-defined transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we will find that the decay of correlations is determined by T r e with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Taking this as a fact for now, we connect the decay of correlations with the spectrum of Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The spectrum of Te depends on k because k determines its dimension and entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Still, we can make general statements about Te for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we will prove the following statements in Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The eigenvalues of Te with e ∈ Sk are non-negative for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ Sk is diagonalizable for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the dis- tinct eigenvalues of Te with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, λ1 = 1 and it is non-degenerate for any k ≥ 2 if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given the statements above, we can expand T r e as T r e = |R1⟩⟨L1| + λr 2 w2 � µ=1 |R(µ) 2 ⟩⟨L(µ) 2 | + O(λr 3), (58) where |R(µ) i ⟩ denotes the µth right eigenvector corre- sponding to λi, ⟨L(µ) i | denotes the µth left eigenvector corresponding to λi, and wi denotes the degeneracy of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The asymptotic decay of correlations is thus deter- mined by the subleading eigenvalue λ2 of Te, and the average correlation length is given by ξ = − 1 log(|λ2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (59) The argument behind this is similar to one known from the analysis of correlations in translation-invariant MPS [56, 65], where the decay is determined by the sub- leading eigenvalue of the relevant transfer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Rényi-2 Mutual Information We start our analysis of correlations in random MPS with the simplest case, namely the computation of the average of the Rényi-2 mutual information I2(A : B) = log � tr � ϱ2 AB �� − log � tr � ϱ2 A �� − log � tr � ϱ2 B �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (60) The analytical treatment turns out to be comparatively simple if one assumes that E log(X) = log(EX), as is frequently done in this context [66–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will not make this assumption further below when we study the von Neumann mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The analysis there will require transfer matrices Tρ with ρ ∈ Sk for all k ≥ 2, while ρ ∈ S2 will suffice here because only the averages of purities are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our first result is summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of I2(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The exact same structure will also appear in the proofs for the other mea- sures of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, this proof serves as the sim- plest example and a point of reference for later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EI2(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, we make the as- sumption that E log(X) = log(EX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, EI2(A : B) = log � E tr � ϱ2 AB �� − log � E tr � ϱ2 A �� − log � E tr � ϱ2 B �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (61) E tr � ϱ2 A � , E tr � ϱ2 B � , and E tr � ϱ2 AB � can already be written in the desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For example, E tr � ϱ2 AB � = tr � PE|ψ⟩⟨ψ|⊗2� (62) with P = � P (d) e �⊗c ⊗ � P (d) (12) �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) (12) �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (63) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EI2(A : B) in terms of the transfer matrices defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given the previous step, it is easy to confirm that EI2(A : B) = log � ⟨I2|T c e T a (12)T r e T b (12)|F2⟩ � − log � ⟨I2|T c e T a (12)|F2⟩ � − log � ⟨I2|T c+a+r e T b (12)|F2⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (64) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EI2(A : B) in terms of the spectrum of Te with e ∈ S2 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (58)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the relevant expressions for the Weingarten function, it is evident [69] that Te = � � � � 1 d2D − D d2D2 − 1 0 dD2 − d d2D2 − 1 � � � � (65) is diagonalizable with λ1 = 1 and λ2 = dD2 − d d2D2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (66) Expanding T c e and taking the limit c → ∞ yields EI2(A : B) = log � ⟨L1|T a (12)T r e T b (12)|F2⟩ � − log � ⟨L1|T a (12)|F2⟩ � − log � ⟨L1|T b (12)|F2⟩ � , (67) where we have used that ⟨I2|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding also T r e and using that |F2⟩ = |R1⟩, we have EI2(A : B) = log � ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + λr 2⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ � − log � ⟨L1|T a (12)|R1⟩ � − log � ⟨L1|T b (12)|R1⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (68) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EI2(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, EI2(A : B) = log � 1 + λr 2 ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ � (69) = λr 2 ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + O � λ2r 2 � (70) ≡ K exp � −r ξ � + O � exp � −2r ξ �� , (71) where K = ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ (72) and ξ = − 1 log(λ2) = − � log � dD2 − d d2D2 − 1 ��−1 = ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (73) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 11 As we discussed earlier, the Rényi-2 mutual informa- tion is lacking many of the desirable properties that a sound measure of correlation ought to fulfill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On top, our computation simplifies considerably because we are using the assumption that E log(X) = log(EX), which amounts to ignoring statistical fluctuations in the dif- ferent realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the following section, we will see that N(A : B) decays exponentially with the same av- erage correlation length ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will furthermore show that N(A : B) concentrates around its average, provid- ing evidence that fluctuations can be safely ignored in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Trace Distance and 2-Norm In this section, we investigate average correlations as quantified by the trace distance T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As anticipated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' II B, this is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, as laid out there, the 2-norm expression N(A : B) reliably esti- mates T(A : B) for the case of random MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Hence, we compute the average of N(A : B) = ∥ϱAB − ϱA ⊗ ϱB∥2 2 (74) = tr � ϱ2 AB � + tr � ϱ2 A � tr � ϱ2 B � − 2 tr[ϱAB(ϱA ⊗ ϱB)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (75) Because of its connection to the Hilbert–Schmidt in- ner product, the average of N(A : B) can be computed without any simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Making use of the transfer-matrix techniques introduced above, we prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of N(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The proof follows the same procedure as that of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Here, we sketch the main steps and refer to App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' H for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Step 1, we write EN(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The second summand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (75) requires permutations of S4 because E tr � ϱ2 A � tr � ϱ2 B � = tr � PE|ψ⟩⟨ψ|⊗4� (76) with P = � P (d) e �⊗c ⊗ � P (d) (12) �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) (34) �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (77) The first summand and the third summand respectively require only permutations of S2 and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Step 2, we thus write EN(A : B) in terms of transfer matrices Tρ with ρ ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This means that the average correlation length is de- termined by the subleading eigenvalue of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Step 3, we find that λ1 = 1 and λ2 = dD2 − d d2D2 − 1, (78) just like for Te with e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The former is non- degenerate, while the degeneracy of the latter is given by the number of transposition in S4, w2 = �4 2 � = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (79) Thus, the average correlation length for N(A : B) co- incides with that for I2(A : B), as we conclude in Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The above result establishes the exponential decay of the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, one is usually interested in know- ing if typical instances are expected to have the same exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This can be easily established by Markov’s inequality because N(A : B) is non-negative and its average decays to zero as a function of the dis- tance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a), sufficiently large r, and all 0 < ε < 1, the random MPS ensemble satisfies Pr � N(A : B) ≥ K exp � −(1 − ε)r ξ1D �� ≤ exp � − εr ξ1D � , (80) where K is constant with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Result 2, for sufficiently large r, we can bound EN(A : B) ≤ K exp(−r/ξ1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because N(A : B) is non-negative, by Markov’s inequality, we have, for η > 0, Pr � N(A : B) ≥ ηK exp � − r ξ1D �� ≤ Pr[N(A : B) ≥ ηEN(A : B)] (81) ≤ 1 η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (82) The result follows with η = exp(εr/ξ1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The above corollary reflects that it is exponentially unlikely in r that N(A : B) decays slower than with the average correlation length ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we have already established that the average case exhibits an exponential decay with correlation length ξ1D, the average case is also typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By combining the above result with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (17), we can now also bound the correlation length for T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='9 ξ 6 8 10 12 14 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='7 d = 2 d = 3 d = 4 d = 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Numerically obtained average correlation length ξ for different d and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The data points are obtained by fitting the average value of I(A : B) against r ∈ {5, 7, 9, 11, 13, 15} for a = b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The sample size of 10 000 suffices for the error bars to lie within the plot points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The opaque curves correspond to ξ1D [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34)] Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a), sufficiently large r, and all 0 < ε < 1, the random MPS ensemble satisfies Pr � T(A : B) ≥ K exp � −(1 − ε)r 2ξ1D �� ≤ exp � − εr ξ1D � , (83) where K is constant with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that ET(A : B) ≤ � E[T(A : B)]2 (84) ≤ D2 2 � EN(A : B) (85) ≤ K exp � − r 2ξ1D � , (86) where, in the last line, we have assumed r to be suffi- ciently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The result follows as in the proof of Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, with overwhelming probability, correlations as quantified by T(A : B) decay exponentially with ξ ≤ 2ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Von Neumann Mutual Information The fact that I2(A : B) and N(A : B) have the same average correlation length ξ1D motivates the question whether other measures of correlation behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In this section, we will provide compelling evidence that ξ1D is indeed the average correlation length also for the von Neumann mutual information I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We start by numerically investigating the behavior of I(A : B) for random MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We have generated MPS ac- cording to our measure, computed the average of I(A : B), and extracted the average correlation length from fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 3 shows, the numerically obtained average correlation length coincides well with ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It should be noted that we have set c = 0 for our numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We discuss in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' J why this does not affect the aver- age correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For more details on our numerical analysis, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We now turn to the analytical computation of the av- erage of I(A : B) = tr[ρAB log(ρAB)] − tr[ρA log(ρA)] − tr[ρB log(ρB)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (87) To be able to make use of the transfer-matrix tech- niques introduced above, we employ two replica tricks to write EI(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First, we write S(ρ) as the α → 1 limit of Sα(ρ): S(ρ) = lim α→1 1 1 − α log[tr(ϱα)] (88) Second, instead of assuming again that E log(X) = log(EX), we use E log(X) = lim v→0 1 v log(EXv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (89) We are thus dealing with expressions of the form tr(PE|ψ⟩⟨ψ|⊗vα), which require transfer matrices Tρ with ρ ∈ Svα (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This means that knowing the spec- trum of Te with e ∈ Svα for all vα ≥ 2 allows us to draw conclusions about the decay of the average of I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While this is in principle a daunting task, we are able to prove several properties of the transfer matrix Te with e ∈ Sk for any k ≥ 2 (see Propositions 1, 2, and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' D, we furthermore show that Te with e ∈ Sk has eigenvalue µ2 = dD2 − d d2D2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (90) for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Its degeneracy is at least v2 = �k 2 � , (91) the number of transpositions in Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We conjecture that µ2 is the subleading eigenvalue of Te with e ∈ Sk for any k ≥ 2 and that it has degeneracy v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We know this conjecture to hold for k ∈ {2, 3, 4}, and we have numerical evidence suggesting so for k ∈ {5, 6, 7} [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We were not able to prove the statement outright, but in the following we argue that it is the only missing step to show that ξ1D is the average correlation length for I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let us state the conjecture formally below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 13 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the dis- tinct eigenvalues of Te with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, λ2 = µ2 with degeneracy w2 = v2 for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, the properties of Te with e ∈ Sva that are relevant for determining the decay of correlations are independent of vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I, we argue that the replica limit does not affect this and prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, the average of I(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We provide a concentration result also for I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement and its proof are identical to that for N(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is exponentially unlikely in r that I(A : B) decays slower than with the average correlation length ξ1D, and the average case is also typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, for subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a), sufficiently large r, and all 0 < ε < 1, the random MPS ensemble satisfies Pr � I(A : B) ≥ K exp � −(1 − ε)r ξ �� ≤ exp � −εr ξ � , (92) where K is constant with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Another corollary of Result 3 is that ξ1D is also the average correlation length for Iα(A : B) for any integer value of α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We prove also this statement in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, for any integer value of α ≥ 1, the average of Iα(A : B) with respect to the ran- dom MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Defini- tion 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, let us summarize the reason behind the per- sistent appearance of the average correlation length ξ1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In all of the examined cases, the asymptotic behavior of the correlations was determined by the asymptotic de- cay of T r e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Although the transfer matrix Te with e ∈ Sk does depend on the number of replicas k, its asymptotic decay does not (given Conjecture 1), resulting in a com- mon average correlation length ξ1D across measures of correlation with different complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' CORRELATIONS IN TWO DIMENSIONS In this section, we state and discuss in more detail the results for random isoTNS summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V A, we develop the two-dimensional analog to the transfer matrices introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A, the tool behind our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V C, we compute the average FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We investigate average correlations in random isoTNS between two subsystems A and B as a function of their hori- zontal distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A and B respectively stretch across a and b consecutive horizontal sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition to indicating the origin of the sequential generation, the diamond also indicates the orthogonality center of the isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (a) For Results 4 and 5, we consider A and B that touch the orthogonality hyper- surface and stretch across h consecutive vertical sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (b) We will provide an additional result for the 2-norm expression N(A : B) for arbitrary (but fixed) A and B that do not need to touch the orthogonality hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' of I2(A : B), and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V D, we investigate the decays of N(A : B) and T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we prove our statements in the limit c → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show the exponential decay of the average for each considered measure of correlation and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) as a function of the distance r between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we prove that I2(A : B) and N(A : B) have a common average correlation length that is independent of the sizes of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moreover, thanks to the more amenable properties of N(A : B), we are able to show that average correlations decay expo- nentially also for subsystems A and B that do not have to touch the orthogonality hypersurface [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Transfer Matrices As in one dimension, computing the average of each measure of correlation will involve computing multiple 14 terms of the form tr � PE|ψ⟩⟨ψ|⊗k� , (93) where P has a similar tensor product structure as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (41), adapted to the two-dimensional setting con- sidered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The number of required replicas k again depends on the considered measure of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will thus need a two-dimensional analog to the transfer matrices introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In contrast to the one-dimensional case, the size of the resulting transfer matrices will also depend on the geometry of the consid- ered subsystems, making their analysis much more chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, the procedure of defining the tensors is similar to that in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We provide an overview here and refer to App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' L for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define V (i,j) = U (i,j) ⊗ U (i,j), where U (i,j) ∈ U � dD2� is the unitary matrix depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By computing the k-fold twirl [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (32)], we obtain the building block = � dU (94) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (95) As in one dimension, it is convenient to define a build- ing block for which the contraction of bond (blue) legs is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Analogously to the one-dimensional case, this results in a tensor with only permutation-valued (green) legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' L, the resulting bulk tensor is given via = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (96) For the sake of brevity, the expressions for the boundary tensors are stated in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Computing expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (93) corre- sponds to contracting each S with some P (d) ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The entries of the resulting bulk tensor Tρ ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' are given by = = (97) = � σ∈Sk Wg � στ −1, dD2� d#(σρ)D#(σθ−1)D#(σν−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (98) In our computations, the site in the top right corner belongs to neither A nor B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is thus acted upon by the trivial permutation e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' L, the corresponding tensor is given by Te = |Fk⟩⟨Fk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' L, we furthermore show that a property similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (55) also holds for random isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, tensors corresponding to the trivial permutation e ∈ Sk on the boundary simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For example, = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (99) As for the one-dimensional case, the proper normaliza- tion of E|ψ⟩⟨ψ|⊗k follows directly from this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Instead of thinking in terms of contractions of two- dimensional tensor networks, it will later prove beneficial to think again in terms of multiplications of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, for any height h of the subsystems A and B [see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a)], we define = (100) 15 as well as = and = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (101) For subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a), we can now write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (93) in terms of transfer matrices: tr � PE|ψ⟩⟨ψ|⊗k� = ⟨Ik|T c e T a α T r e T b β |Fk⟩ (102) We provide an additional Mathematica package [63] that defines Tρ with ρ ∈ Sk for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , 20} according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Once again, the package relies on the one provided by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [64] for evaluating the Weingarten function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Estimating the Decay of Correlations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (102) implies that, for subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a), the decay of the average of each measure of correlation will again reduce to a statement in terms of transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, the decay will be determined by the spectrum of Te with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Notice that, in addition to a dependence on k, the form and properties of Te now depend also on the height h of the subsystems A and B [see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, as we prove in the following sections, its two leading eigen- values are independent of h for at least k = 2 and k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Crucially, this will allow us to make statements about the decay of the averages of I2(A : B) and N(A : B) for arbitrary h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Note that we could, in principle, investigate vertical separation instead of horizontal separation because = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (103) The underlying exchange of indices does not affect the spectrum of the relevant identity transfer matrix and thus neither the average correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This reflects the fact that the sequential generation procedure is symmet- ric in the horizontal and vertical spatial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Rényi-2 Mutual Information In this section, we compute the average of the Rényi-2 mutual information I2(A : B) for random isoTNS that are generated as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 (2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Subsystems A and B are defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we will make the assumption that E log(X) = log(EX) (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1 of the proof of Result 4 (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' O) is thus all but identical to Step 1 of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Also Step 2 is largely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To see this, let us take E tr � ϱ2 A � as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With A and B as defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) and x = (12), E tr � ϱ2 A � = (104) = (105) = ⟨I2|T c e T a x |F2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (106) The resulting expression for EI2(A : B), EI2(A : B) = log � ⟨I2|T c e T a x T r e T b x |F2⟩ � − log � ⟨I2|T c e T a x |F2⟩ � − log � ⟨I2|T c+a+r e T b x |F2⟩ � , (107) resembles Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (64) closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The remaining technical challenge is the analysis of the spectrum of Te with e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote its distinct eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M that λ1 = 1 and λ2 = dD3 − dD d2D4 − 1 (108) and that both eigenvalues are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the proof, that holds for any h, we map the contraction of tensors defining Te with e ∈ S2 to a multiplication of ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the Weingarten calculus, we show that Te is upper block triangular, a property that simplifies the analysis of its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The main difficulty is then to prove that the specified λ2 is indeed the subleading eigen- value for all h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We do this by exploiting substochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Following the same reasoning as before, we find that the average of I2(A : B) decays exponentially as specified in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We prove this result in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of I2(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Trace Distance and 2-Norm In this section, we show the exponential decay of the average of N(A : B) for random isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As for the one- dimensional case, we do that to eventually make conclu- sions about the behavior of the trace distance T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While it is not trivial to compute the average of N(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a), the computation follows along the lines of what we have laid out in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we find that the decay of the average of N(A : B) is determined by the spectrum of the transfer matrix Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' N, for any h, λ1 = 1 and λ2 = dD3 − dD d2D4 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (109) As in one dimension, the former is non-degenerate, while the degeneracy of the latter is given by the number of transpositions in S4, w2 = �4 2 � = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (110) While the analysis of the spectrum is, in principle, similar to that of the spectrum of Te with e ∈ S2, it is consider- ably more technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This is largely due to the fact that the matrices whose multiplication defines Te with e ∈ S4 are significantly more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Still, we find also Te with e ∈ S4 to be upper block triangular, allowing us to show that the specified λ2 is indeed the subleading eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given λ2 of Te with e ∈ S4 and the arguments devel- oped in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV D, we can state the first result of this section, which we prove in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Result 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of N(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We now turn to the case of correlations in isoTNS with arbitrary (but fixed) subsystems A and B [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The decay of the average of N(A : B) for arbitrary A and B can be bounded by employing the fact that the Schatten 2-norm satisfies [70] ∥trB(XAB)∥2 ≤ � dim(B)∥XAB∥2, (111) where XAB is any bipartite operator and dim(B) is the dimension of the Hilbert space that is traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This means that N(A : B) ≤ d|AC|+|BC|N(A′ : B′), (112) where A is now an arbitrary subsystem, A′ is its (min- imal) enclosing rectangle that touches the hypersurface, and AC = A′ − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' B′ and BC are defined analogously for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the statement above, we can bound the decay of the average of N(A : B) for arbitrary A and B as a straightforward corollary of Result 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We stress that we consider regime in which the distance r between A and B grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For arbitrary subsystems A and B, the average of N(A : B) with respect to the random isoTNS ensemble decays as N(A : B) = O � exp � − r ξ2D �� , (113) where the average correlation length ξ2D is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we state a concentration result for N(A : B), which, in combination with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (16), also allows us to draw conclusions about the typical behavior of T(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As before, we consider arbitrary (but fixed) subsystems A and B [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For arbitrary subsystems A and B, suffi- ciently large r, and all 0 < ε < 1, the random isoTNS ensemble satisfies Pr � N(A : B) ≥ K exp � −(1 − ε)r ξ2D �� ≤ exp � − εr ξ2D � , (114) where K is constant with respect to r and the average correlation length ξ2D is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For arbitrary subsystems A and B, suffi- ciently large r, and all 0 < ε < 1, the random isoTNS ensemble satisfies Pr � T(A : B) ≥ K exp � −(1 − ε)r 2ξ2D �� ≤ exp � − εr ξ2D � (115) where K is constant with respect to r and the average correlation length ξ2D is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The proofs and discussions of these results are identical to their one-dimensional counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The tools we have developed in this and the previous two sections should, in principle, allow us to make state- ments also about the decay of the average of the von Neumann mutual information I(A : B) with respect to the random isoTNS ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we would need to investigate the spectrum of Te with e ∈ Sk for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, as the analysis of the spectrum of Te is already quite technical for e ∈ S4 with our methods, we refrain from tackling the spectrum for k ≥ 5 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' CONCLUSION We have investigated the average behavior of correla- tions between two distant subsystems A and B for en- sembles of random MPS and isoTNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As measures of 17 correlation, we have considered the Rényi-α mutual in- formation, a measure arising from the Hilbert–Schmidt norm, the trace distance, and the von Neumann mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We have shown that the average of each considered measure exhibits an exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Our results can equivalently be seen as describing states re- sulting from quantum circuits with a sequential architec- ture and Haar random gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By leveraging the Weingarten calculus, we have devel- oped a mathematical framework that allows to infer the average correlation length from the subleading eigenvalue of an appropriately defined transfer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We have computed the averages of the Rényi-α mutual informa- tion and the measure arising from the Hilbert–Schmidt norm to show the emergence of an average correlation length that only depends on the underlying spatial di- mension but not the considered measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, the average correlation length for random MPS increases weakly with the bond dimension D and converges rapidly (as D grows) to the value 1/ log(d), where d is the phys- ical dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' On the contrary, the average correlation length for random isoTNS, while still depending on the bond dimension only weakly, decreases with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Surpris- ingly, the highest average correlation length for random isoTNS is achieved with the lowest non-trivial bond di- mension (D = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using elementary concentration results, we have fur- thermore deduced the typical behavior of the measure arising from the Hilbert–Schmidt, which has in turn al- lowed us to make similar statements about the trace dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For MPS, we have been able to give strong indications that the universal correlation length applies also to the von Neumann mutual information, and also any Rényi- α mutual information for integer values of α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It would be interesting to prove this behavior rigorously, and also investigate its validity for the isoTNS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' An- other possible future direction would be to study average correlations in more general random PEPS, beyond the class of isoTNS, as well as other types of quantum circuit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Ignacio Cirac and Philippe Faist for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' and G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [75] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Sanz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Perez-Garcia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Wolf, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Cirac, A Quantum Version of Wielandt’s inequality, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Theory 56, 4668 (2010), arXiv:0909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='5347 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [76] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Collins and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Matsumoto, Weingarten calculus via orthogonality relations: new applications, Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 14, 631 (2017), arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='04493 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [77] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Akers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Faulkner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Rath, Reflected entropy in random tensor networks, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2022, 162, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='09122 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [78] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Biane, Some properties of crossings and partitions, Discrete Mathematics 175, 41 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [79] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Castellani and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Cavagna, Spin-glass theory for pedestrians, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2005, P05012 (2005), arXiv:cond-mat/0505032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix A: k-Fold Twirl In this appendix, we present some additional details on the k-fold twirl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we go from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (18) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To do this, we will need a result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [54], which appears as Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='4 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We state it without proof as Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let k be a positive integer, and (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , ik), (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , jk), (ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , ℓk), and (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , mk) be k-tuples of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, � dU Ui1ji · · · UikjkUm1ℓ1 · · · Umkℓk = � σ,τ∈Sk Wg � σ−1τ, q � δi1mσ(1) · · · δikmσ(k)δj1ℓτ(1) · · · δjklτ(k), (A1) where the integration is with respect to the Haar measure on the unitary group U(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 20 By Lemma 1, � T (k) U (X) � i1···ikm1···mk = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',ℓk � dU Ui1ji · · · UikjkXj1···jkℓ1···ℓkUm1ℓ1 · · · Umkℓk (A2) = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',ℓk � σ,τ∈Sk Wg � σ−1τ, q � δi1mσ(1) · · · δikmσ(k)Xj1···jkℓ1···ℓkδj1ℓτ(1) · · · δjklτ(k) (A3) = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk � σ,τ∈Sk Wg � σ−1τ, q � δi1mσ(1) · · · δikmσ(k)Xj1···jkjτ−1(1)···jτ−1(k) (A4) = � σ,τ∈Sk Wg � σ−1τ, q �� P (q) σ−1 � i1···ikm1···mk tr � XP (q) τ � , (A5) where, in the final line, we have used that � P (q) σ−1 � i1···ikm1···mk = ⟨i1 · · · ik|P (q) σ−1|m1 · · · mk⟩ (A6) = ⟨i1 · · · ik|mσ(1) · · · mσ(k)⟩ (A7) = δi1mσ(1) · · · δikmσ(k) (A8) and that tr � XP (q) τ � = ��� j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk ⟨j1 · · · jk|XP (q) τ |j1 · · · jk⟩ (A9) = � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',jk ⟨j1 · · · jk|X|jτ −1(1) · · · jτ −1(k)⟩ (A10) = Xj1···jkjτ−1(1)···jτ−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (A11) Thus, T (k) U (X) = � σ,τ∈Sk Wg � σ−1τ, q � P (q) σ−1 tr � XP (q) τ � (A12) = � σ,τ∈Sk Wg � στ −1, q � P (q) σ tr � XP (q) τ −1 � (A13) = � σ,τ∈Sk Wg � στ −1, q � P (q) σ tr � X � P (q) τ �T � , (A14) which coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In addition, let us confirm that T (k) U � P (q) ρ � = P (q) ρ [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Indeed, T (k) U � P (q) ρ � = � σ,τ∈Sk Wg � στ −1, q � P (q) σ tr � P (q) ρ � P (q) τ �T � (A15) = � σ,τ∈Sk Wg � στ −1, q � P (q) σ q#(ρτ −1) (A16) = � σ∈Sk δρσP (q) σ (A17) = P (q) ρ , (A18) where, in the third line, we have used the definition of the Weingarten function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix B: Proofs of Propositions 1 and 2 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The eigenvalues of Te with e ∈ Sk are non-negative for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 21 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ Sk is diagonalizable for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To prove Propositions 1 and 2, we will define matrices X ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' and Y ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' so that Ck = WXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will then discuss some properties of those three matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The proofs themselves will boil down to similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define the diagonal matrix X ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' via Xσσ = d#(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (B1) It is evident that X is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define the Gram matrix Y ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' via Yσθ = tr � P (D) σ � P (D) θ �T � = D#(σθ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (B2) Y is positive semidefinite because it is a Gram matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The Weingarten matrix Wg � στ −1, q � = � G−1� στ is positive definite because the Gram matrix G [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (21)] is positive definite [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will need the fact that Y W is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Y W has non-negative eigenvalues because it is a product of a positive semidefinite (Y ) and a positive definite matrix (W) (see Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Furthermore, Y W is symmetric: ⟨τ|Y W|θ⟩ = � σ∈Sk Wg � στ −1, dD � D#(σθ−1) (B3) = � π∈Sk Wg � πθτ −1, dD � D#(π) (B4) = � π∈Sk Wg � τθ−1π−1, dD � D#(π) (B5) = � π∈Sk Wg � θ−1π−1τ, dD � D#(π) (B6) = � ϕ∈Sk Wg � θ−1ϕ, dD � D#(τϕ−1) (B7) = � ϕ∈Sk Wg � ϕθ−1, dD � D#(ϕτ −1) (B8) = ⟨θ|Y W|τ⟩ (B9) In the third line, we have used that Wg(α, q) = Wg � α−1, q � , and, in the fourth line, we have used that Wg(α, q) = Wg � βαβ−1, q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Both identities are a result of the Weingarten function being sensitive only to the conjugacy class of a given permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Ck = WXY is similar to XY W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A product of a positive definite (X) and a positive semidef- inite matrix (Y W), XY W has non-negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows by similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Ck = WXY is similar to XY W, which is similar to X1/2Y WX1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because Y W is symmetric, so is X1/2Y WX1/2, which makes the latter diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix C: Proof of Proposition 3 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, λ1 = 1 and it is non-degenerate for any k ≥ 2 if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For what follows, it is convenient to define the transfer matrix Σe with e ∈ Sk via = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C1) 22 Our strategy will be to prove the claimed spectral property for Σe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This is enough because, for any two operators X and Y for which XY and Y X are well defined, the sets of eigenvalues of XY and Y X are the same (up to zeros and the multiplicity of the non-zero eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (51) and (C1), Tρ and Σρ are related in exactly this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As Σe arises from the contraction of the quantum channel underlying R [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (44)] with the identity permutation, it can be understood as a generalization of the k-fold twirling operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, it involves an ”environment” E of dimension � Cd�⊗k that is eventually traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As a superoperator (that is, without using the operator-vector correspondence), it reads [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (18)] Σe(·) = � dU trE � U ⊗k � k � l=1 |0⟩⟨0|El ⊗ (·)Sl � � U †�⊗k � , (C2) where the integration is with respect to the Haar measure on the unitary group U(dD), E = �k l=1 El corresponds to � Cd�⊗k, and S = �k l=1 Sl corresponds to � CD�⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is apparent that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C2) represents a convex combination of quantum channels (note the Stinespring dilation form), and thus the resulting operator is also a valid quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This implies that 1 is an eigenvalue of Σe and that there is no eigenvalue of greater modulus [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We now prove that no other eigenvalue of the same modulus exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, we will show that Σe is a primitive channel [74], which implies said property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' A quantum channel is primitive if and only if the spanning space formed by products of its Kraus operators, Km = span �� m � k=1 Kik �� , (C3) is equal to the full matrix algebra for some integer m, that is, Km = MDk(C) [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show that this condition is satisfied for Σe if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Indeed, the Haar integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C2), together with the partial trace, can be understood as a (redundant) Kraus decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Precisely, we can take {Ki}i = � trE �� k � l=1 |+⟩⟨ψl|El ⊗ ISl � U ⊗k �� U,ψ , (C4) where U ∈ U(dD), |ψl⟩ ∈ {|0⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , |d − 1⟩} is the computational basis of the lth replica of the environment, and |+⟩ = �d−1 j=0 |j⟩/ √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The latter is a choice (instead of |0⟩) made for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It remains to show that there exists an integer m = m(k, d, D) such that Km = MDk(C) if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First of all, note that the above fails for d = 1 (that is, if the environment E is trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This is because span �� U ⊗k� U � coincides with the symmetric subspace over the k subsystems [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' However, d = 2 is already enough to span the full matrix algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' An explicit construction to show this fact amounts to taking U to be a controlled unitary gate, where the control system is E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, consider |ψl⟩ = |δlr⟩ and U = |0⟩⟨0|E ⊗ IS + d−1 � j=1 |j⟩⟨j|E ⊗ VS, (C5) where r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k}, and VS is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This results in Kraus operators [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C4)] of the form IS1 ⊗ · · · ⊗ ISr−1 ⊗ V ⊗ ISr+1 · · · ⊗ ISk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C6) Taking (finite) products, as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (C3) dictates, is enough to build a basis of the vector space MDk(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Note that this construction requires two control levels (that is, d ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix D: Further Properties of the Transfer Matrix Te In this appendix, we state and prove statements about the structure of the transfer matrix Te with e ∈ Sk that will motivate Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We prove the statements in Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' E, F, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Moving forward, we denote Te with e ∈ Sk by Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each entry ⟨τ|Ck|θ⟩ = � σ∈Sk Wg � στ −1, dD � d#(σ)D#(σθ−1) (D1) 23 of Ck ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='×k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' is a sum of k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While this may sound daunting, Ck exhibits a structure that reduces its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As formalized by Proposition 4, the entries ⟨τ|Ck|θ⟩ of Ck exhibit a dependence on the conjugacy class of their indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, if we know the entries of the column given by θ ∈ Sk, we also know the entries of the columns given by permutations in the same conjugacy class as θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any π ∈ Sk, ⟨τ|Ck|θ⟩ = ⟨πτπ−1|Ck|πθπ−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D2) Certain entries of Ck vanish, while others are given by the entries of Ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 5 captures these two statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For all θ ∈ Sk with θ(k) = k, ⟨τ|Ck|θ⟩ = δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, (D3) where ρ↓ ∈ Sk−1 is the restriction of ρ ∈ Sk with ρ(k) = k to the permutation on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To understand the strength of Propositions 4 and 5, let us have a look at C2 and C3, the two most simple transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With bases S2 = {e, (12)} and S3 = {e, (12), (13), (23), (123), (132)}, one finds that C2 = � 1 α 0 β � and C3 = � � � � � � � 1 α α α γ γ 0 β 0 0 δ δ 0 0 β 0 δ δ 0 0 0 β δ δ 0 0 0 0 ε ζ 0 0 0 0 ζ ε � � � � � � � , (D4) where each Greek letter corresponds to some function of d and D whose exact form is not relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The entries in the first four columns of C3 are fully determined by those of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The entries in the last two columns of C3 do not arise from those of C2, but the sixth column is a permutation of the fifth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Note that we are deliberately choosing a basis Sk = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , sk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='} that makes the special structure of Ck more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we sort permutations so that those with i fixed points come before those with i − 1 fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We group permutations that have common fixed points and then those that are in the the same conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given this basis, Propositions 4 and 5 imply that Ck is block triangular with k diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We denote by C(1) k the diagonal block with τ = θ = e ∈ Sk and by C(i) k with 2 ≤ i ≤ k the diagonal block corresponding to τ, θ ∈ Sk with k − i fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The spectrum of Ck is then given by λ(Ck) = λ � C(1) k � ∪ λ � C(2) k � ∪ · · · ∪ λ � C(k) k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D5) It is apparent that C(1) k has a single, non-degenerate eigenvalue µ1 = ⟨e|Ck|e⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D6) C(2) k has a single, degenerate eigenvalue µ2 = ⟨(12)|Ck|(12)⟩ = dD2 − d d2D2 − 1, (D7) which corresponds to the expression of β in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The degeneracy of µ2 is given by the size of the block, which is in turn given by the number of transpositions in Sk, v2 = �k 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D8) By Proposition 3, 1 is the leading eigenvalue of Ck and non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We conjecture that µ2 is the subleading eigenvalue of Ck and that it has degeneracy v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement of Conjecture 1 holds for k ∈ {2, 3, 4}, and we have numerical evidence suggesting so for k ∈ {5, 6, 7} [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 24 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, λ2 = µ2 with degeneracy w2 = v2 for any k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As formalized by Proposition 6, the statements above hold for any transfer matrix Tρ with ρ ∈ Sk because Tρ is similar to Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any ρ ∈ Sk, Tρ = QT ρ CkQρ with Qρ = � π∈Sk |ρπ⟩⟨π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D9) Appendix E: Proof of Proposition 4 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any π ∈ Sk, ⟨τ|Ck|θ⟩ = ⟨πτπ−1|Ck|πθπ−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that ⟨τ|Ck|θ⟩ = � σ∈Sk Wg � στ −1, dD � d#(σ)D#(σθ−1) (E1) = � ϕ∈Sk Wg � π−1ϕπτ −1, dD � d#(π−1ϕπ)D#(π−1ϕπθ−1) (E2) = � ϕ∈Sk Wg � ϕπτ −1π−1, dD � d#(π−1ϕπ)D#(ϕπθ−1π−1) (E3) = � ϕ∈Sk Wg � ϕ � πτπ−1�−1, dD � d#(ϕ)D # � ϕ(πθπ−1) −1� (E4) = ⟨πτπ−1|Ck|πθπ−1⟩, (E5) where, in the second line, we have used that the conjugation map is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the third line, we have used that Wg(α, q) = Wg � βαβ−1, q � and that #(α) = # � βαβ−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Both identities are a result of the two functions being sensitive only to the conjugacy class of a given permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix F: Proof of Proposition 5 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For all θ ∈ Sk with θ(k) = k, ⟨τ|Ck|θ⟩ = δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, (D3) where ρ↓ ∈ Sk−1 is the restriction of ρ ∈ Sk with ρ(k) = k to the permutation on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To prove Proposition 5, we will need a number of ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The main one will be Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [76], which we state without proof as Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any ρ ∈ Sk, k−1 � i=1 Wg � (ik)π, q � + q Wg(π, q) = δk,π(k) Wg � π↓, q � , (F1) where (ik) denotes the transposition of elements i and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To use Lemma 2, we must do two things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First, we must split the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D1) into certain sums of k terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We employ Lemma 3 to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any α ∈ Sk, there exists a β ∈ Sk with β(k) = k such that either α = β or α = (ik)β with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If α(k) = k, then α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Otherwise, k is in exactly one of the disjoint cycles of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Without loss of generality, say α = (kia3a4 · · · )(b1b2 · · · ) · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, α = (ik)(ia3a4 · · · )(b1b2 · · · ) · · · ≡ (ik)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Second, we must ensure that summands in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D1) with Wg(π, dD) have a factor dD while those with Wg � (ik)π, dD � do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We achieve this with Lemma 5 whose proof employs Lemma 4 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [77], which we state without proof as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The lemma also appears as Lemma 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let α ∈ Sk be a transposition, and β ∈ Sk such that #(β) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If the elements exchanged by α are not in the same cycle of β, then #(αβ) = u − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' # � ϕθ−1� = # � (ik)ϕθ−1� + 1 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' # � ϕθ−1� − #(ϕ) = # � (ik)ϕθ−1� − # � (ik)ϕ � for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, � ϕθ−1� (k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, k is in a cycle by itself and thus not in the same cycle of ϕθ−1 as i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let ϕ, θ ∈ Sk with ϕ(k) = k and θ(k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, i and k are not in the same cycle of neither ϕθ−1 nor ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Lemma 4, # � ϕθ−1� = # � (ik)ϕθ−1� + 1 and #(ϕ) = # � (ik)ϕ � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, # � ϕθ−1� − #(ϕ) = # � (ik)ϕθ−1� + 1 − # � (ik)ϕ � − 1 = � (ik)ϕθ−1� − # � (ik)ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With that, we can prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Lemma 3, ⟨τ|Ck|θ⟩ = � σ∈Sk Wg � στ −1, dD � d#(σ)D#(σθ−1) (F2) = � ϕ∈Sk ϕ(k)=k �k−1 � i=1 Wg � (ik)ϕτ −1, dD � d# � (ik)ϕ � D#((ik)ϕθ−1) + Wg � ϕτ −1, dD � d#(ϕ)D#(ϕθ−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (F3) By Lemma 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' for θ ∈ Sk with θ(k) = k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ⟨τ|Ck|θ⟩ = � ϕ∈Sk ϕ(k)=k 1 d#(ϕθ−1)−#(ϕ) �k−1 � i=1 Wg � (ik)ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � (dD)#((ik)ϕθ−1) + Wg � ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � (dD)#(ϕθ−1) � (F4) = � ϕ∈Sk ϕ(k)=k (dD)#(ϕθ−1)−1 d#(ϕθ−1)−#(ϕ) �k−1 � i=1 Wg � (ik)ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � + dD Wg � ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � � (F5) = � ϕ∈Sk ϕ(k)=k d#(ϕ)−1D#(ϕθ−1)−1 �k−1 � i=1 Wg � (ik)ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � + dD Wg � ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � � (F6) = � ϕ∈Sk ϕ(k)=k d#(ϕ↓)D # � (ϕθ−1] ↓��k−1 � i=1 Wg � (ik)ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � + dD Wg � ϕτ −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' dD � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (F7) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' in the final line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' we have used that # � α↓� = #(α) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By Lemma 2, ⟨τ|Ck|θ⟩ = � ϕ∈Sk ϕ(k)=k δk,(ϕτ −1)(k) Wg �� ϕτ −1�↓, dD � d#(ϕ↓)D # � (ϕθ−1) ↓� (F8) = δk,τ(k) � ϕ∈Sk ϕ(k)=k Wg �� ϕτ −1�↓, dD � d#(ϕ↓)D # � (ϕθ−1) ↓� (F9) = δk,τ(k)⟨τ ↓|Ck−1|θ↓⟩, (F10) 26 where, in the second line, we have used that � ϕτ −1� (k) = k if and only if τ(k) = k because ϕ(k) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix G: Proof of Proposition 6 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For any ρ ∈ Sk, Tρ = QT ρ CkQρ with Qρ = � π∈Sk |ρπ⟩⟨π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (D9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that ⟨τ|Tρ|θ⟩ = � σ∈Sk Wg � στ −1, dD � d#(σρ)D#(σθ−1) (G1) = � σ∈Sk Wg � ρστ −1ρ−1, dD � d#(ρσ)D#(ρσθ−1ρ−1) (G2) = � σ∈Sk Wg � ρσ(ρτ)−1, dD � d#(ρσ)D#[ρσ(ρθ)−1] (G3) = � π∈Sk Wg � π(ρτ)−1, dD � d#(π)D#[π(ρθ)−1] (G4) = ⟨ρτ|Ck|ρθ⟩, (G5) where, in the second line, we have used that Wg(α, q) = Wg � βαβ−1, q � and that #(α) = # � βαβ−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Both identities are a result of the two functions being sensitive only to the conjugacy class of a given permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the fourth line, we have used that the left-multiplication map is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix H: Proof of Result 2 Result 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of N(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps, following the structure of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EN(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With the Hilbert-Schmidt inner product, EN(A : B) = E tr � ϱ2 AB � + E tr � ϱ2 A � tr � ϱ2 B � − 2E tr[ϱAB(ϱA ⊗ ϱB)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(H1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='It is then easy to confirm that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='EN(A : B) = tr ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EN(A : B) in terms of the transfer matrices defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given the previous step, it is easy to confirm that EN(A : B) = ⟨I4|T c e T a (12)T r e T b (12)|F4⟩ + ⟨I4|T c e T a (34)T r e T b (12)|F4⟩ − 2⟨I4|T c e T a (12)T r e T b (13)|F4⟩ (H3) = ⟨I4|T c e T a (12)T r e � T b (12) + T b (34) − 2T b (13) � |F4⟩ (H4) ≡ ⟨I4|T c e AT r e B|F4⟩, (H5) where we have defined A = T a (12) and B = T b (12) + T b (34) − 2T b (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H6) 27 Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EN(A : B) in terms of the spectrum of Te with e ∈ S4 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (58)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let λ1 > λ2 > · · · ≥ 0 denote the distinct eigenvalues of Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds [63] that λ1 = 1 and λ2 = dD2 − d d2D2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H7) The former is non-degenerate, while the degeneracy of the latter is given by the number of transposition in S4 [63], w2 = �4 2 � = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H8) Expanding T c e and taking the limit c → ∞ yields EN(A : B) = ⟨L1|AT r e B|F4⟩, (H9) where we have used that ⟨I4|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding T r e and using that |F4⟩ = |R1⟩, we have EN(A : B) = ⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3) (H10) = λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3), (H11) where, in the second line, we have used that ⟨L1|B|R1⟩ = 0, (H12) which we prove in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we prove that ⟨L1|T b t |R1⟩ does not depend on the two elements the transposition t ∈ S4 acts upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We need some understanding of T b t as well as the eigenvectors ⟨L1| and |R1⟩ of Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For the former, we make use of the fact that T b ρ with ρ ∈ S4 is similar to to T b e with e ∈ S4, T b ρ = � π,ϕ∈S4 |π⟩⟨ρπ|Cb 4|ρϕ⟩⟨ϕ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H13) With α = d2D − D d2D2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' β = dD2 − d d2D2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' f(u) = u−1 � i=0 αβi and g(u) = βu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(H14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='Te|si⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H15) For getting some understanding of ⟨L1|, we make use of the fact that ⟨L(µ) i |R(ν) j ⟩ = δijδµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is easy to confirm that |R1⟩ = |s1⟩ and |R(µ) 2 ⟩ = α β − 1|s1⟩ + |sµ+1⟩, (H16) which implies that � ⟨L1|si⟩ � 1≤i≤7 = � 1 α β − 1 α β − 1 α β − 1 α β − 1 α β − 1 α β − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H17) For any transposition t ∈ S4, it thus holds that ⟨L1|T b t |R1⟩ = � π,ϕ∈S4 ⟨L1|π⟩⟨tπ|Cb 4|tϕ⟩⟨ϕ|R1⟩ (H18) = � π,ϕ∈S4 ⟨L1|π⟩⟨tπ|Cb 4|tϕ⟩δs1ϕ (H19) = � π∈S4 ⟨L1|π⟩⟨tπ|Cb 4|t⟩ (H20) = � π∈S4 f(b)⟨L1|π⟩⟨tπ|s1⟩ + � π∈S4 g(b)⟨L1|π⟩⟨tπ|t⟩ (H21) = � π∈S4 f(b)⟨L1|π⟩δtπ + � π∈S4 g(b)⟨L1|π⟩δs1π (H22) = f(b)⟨L1|t⟩ + g(b)⟨L1|s1⟩ (H23) = α β − 1f(b) + g(b), (H24) which is independent of the two elements the transposition t ∈ S4 acts upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Thus, ⟨L1|B|R1⟩ = ⟨L1| � T b (12) + T b (34) − 2T b (13) � |R1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H25) 29 Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EN(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, EN(A : B) ≡ K exp � −r ξ � + O � exp � − r χ �� , (H26) where K = w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ (H27) and ξ = − 1 log(λ2) = − � log � dD2 − d d2D2 − 1 ��−1 = ξ1D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H28) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix I: Proof of Result 3 and Corollary 4 Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, the average of I(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' If Conjecture 1 holds, for any integer value of α ≥ 1, the average of Iα(A : B) with respect to the random MPS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 2 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ1D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In this appendix, we prove Result 3 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The proof of the latter will follow directly from the proof of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps, following the structure of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because I(A : B) and I2(A : B) are related, the steps are overall very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As is usual in the context of the replica trick [79], we will interchange the order of some limits without rigorous justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EI(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' To that end, we make use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (88) and (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With those, EI(A : B) = lim α→1 lim v→0 1 vα − v � log � E tr(ϱα AB)v� − log � E tr(ϱα A)v� − log � E tr(ϱα B)v�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I1) E tr(ϱα A)v, E tr(ϱα A)v, and E tr(ϱα AB)v can be written in the desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let us define x ∈ Sα to be the cyclic permutation so that x(i) = i + 1 modulo α and xw = � α(w − 1) + 1, α(w − 1) + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , αw � ∈ Svα (I2) with w ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, for example, E tr(ϱα AB)v = tr ��� P (d) e �⊗c ⊗ � P (d) x �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) x �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e � E|ψ⟩⟨ψ|⊗α �v (I3) = tr ��� P (d) e �⊗c ⊗ � P (d) x1···xv �⊗a ⊗ � P (d) e �⊗r ⊗ � P (d) x1···xv �⊗b ⊗ � P (d) e �⊗(f−1) ⊗ P (dD) e � E|ψ⟩⟨ψ|⊗vα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I4) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EI(A : B) in terms of the transfer matrices defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Given the previous step, it is easy to confirm that EI(A : B) = lim α→1 lim v→0 1 vα − v � log � ⟨Ivα|T c e T a x1···xvT r e T b x1···xv|Fvα⟩ � − log � ⟨Ivα|T c e T a x1···xv|Fvα⟩ � − log � ⟨Ivα|T c+a+r e T b x1···xv|Fvα⟩ �� (I5) ≡ lim α→1 lim v→0 1 vα − v � log(⟨Ivα|T c e AT r e B|Fvα⟩) − log(⟨Ivα|T c e A|Fvα⟩) − log � ⟨Ivα|T c+a+r e B|Fvα⟩ �� (I6) where we have defined A = T a x1···xv and B = T b x1···xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I7) 30 Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EI(A : B) in terms of the spectrum of Te with e ∈ Svα [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (58)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' At this point, we assume Conjecture 1 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, for any vα ≥ 2, we assume that λ2 = dD2 − d d2D2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I8) with degeneracy w2 = �k 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I9) Expanding T c e and taking the limit c → ∞ yields EI(A : B) = lim α→1 lim v→0 1 vα − v [log(⟨L1|AT r e B|Fvα⟩) − log(⟨L1|A|Fvα⟩) − log(⟨L1|B|Fvα⟩)], (I10) where we have used that ⟨Ivα|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding T r e and using that |Fvα⟩ = |R1⟩, we have EI(A : B) = lim α→1 lim v→0 1 vα − v � log � ⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3) � − log(⟨L1|A|R1⟩) − log(⟨L1|B|R1⟩) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I11) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EI(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With Λ = max �� λ2r 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' λr 3 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' EI(A : B) = lim α→1 lim v→0 1 vα − v log � 1 + λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ ⟨L1|A|R1⟩⟨L1|B|R1⟩ + O(λr 3) � (I12) = lim α→1 lim v→0 1 vα − v � λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ ⟨L1|A|R1⟩⟨L1|B|R1⟩ + O(Λ) � (I13) ≡ lim α→1 lim v→0 1 vα − v � �K(vα) exp � −r ξ � + O � exp � − r χ ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I14) where �K(vα) = w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ ⟨L1|A|R1⟩⟨L1|B|R1⟩ (I15) and ξ = − 1 log(λ2) = − � log � dD2 − d d2D2 − 1 ��−1 = ξ1D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I16) As ξ is independent of vα, it cannot be affected by the replica limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' �K(vα) will converge to some K that is guaranteed to be independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (89), it holds that EIα(A : B) = lim v→0 1 vα − v � log � E tr(ϱα AB)v� − log � E tr(ϱα A)v� − log � E tr(ϱα B)v�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (I17) Thus, the proof is identical to that of Result 3 without the limit α → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 31 Appendix J: Result 3 with c = 0 In this appendix, we prove a version of Result 3 with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Steps 1 and 2 of this proof are identical to Steps 1 and 2 of the proof of Result 3 with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, at the end of Step 2, we have EI(A : B) = lim α→1 lim v→0 1 vα − v � log(⟨Ivα|ACr vαB|Fvα⟩) − log(⟨Ivα|A|Fvα⟩) − log � ⟨Ivα|Ca+r vα B|Fvα⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (J1) We start the proof at Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EI(A : B) in terms of the spectrum of Te with e ∈ Svα [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (58)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We assume Conjecture 1 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Expanding T r e and using that ⟨Ivα|R1⟩ = 1 yields EI(A : B) = lim α→1 lim v→0 1 vα − v � log � ⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ + λr 2 w2 � µ=1 ⟨Ivα|A|R(µ) 2 ⟩⟨L(µ) 2 |B|Fvα⟩ + O(λr 3) � − log(⟨Ivα|A|Fvα⟩) − log � ⟨L1|B|Fvα⟩ + λa+r 2 w2 � µ=1 ⟨Ivα|R(µ) 2 ⟩⟨L(µ) 2 |B|Fvα⟩ + O(λr 3) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (J2) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We write EI(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With Λ = max �� λ2r 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' λr 3 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='EI(A : B) = lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='α→1 lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='v→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='vα − v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 + λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ O(λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='− log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 + λa+r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨L1|B|Fvα⟩ + O(λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(J3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='= lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='α→1 lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='v→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='vα − v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ λa+r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨L1|B|Fvα⟩ + O(Λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(J4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='= lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='α→1 lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='v→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='vα − v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='λr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='µ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ λa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2⟨Ivα|R(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 ⟩⟨L(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='2 |B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨L1|B|Fvα⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ O(Λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(J5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='≡ lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='α→1 lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='v→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='vα − v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='K′(vα) exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='− r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='χ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (J6) where � K′(vα) = w2 � µ=1 � ⟨Ivα|A|R(µ) 2 ⟩⟨L(µ) 2 |B|Fvα⟩ ⟨Ivα|A|R1⟩⟨L1|B|Fvα⟩ + λa 2⟨Ivα|R(µ) 2 ⟩⟨L(µ) 2 |B|Fvα⟩ ⟨L1|B|Fvα⟩ � (J7) and ξ = − 1 log(λ2) = − � log � dD2 − d d2D2 − 1 ��−1 ξ1D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (J8) Again, � K′(vα) will converge to some K′ that is guaranteed to be independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While K′ is different from K in general, the correlation length ξ is independent of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix K: Numerical Analysis In this appendix, we briefly review our numerical analysis of the von Neumann mutual information I(A : B) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' IV E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We fix d and D, and we set a = b = 1 and r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (i) We generate a + r + b + 1 Haar-random unitary matrices of U(dD) to define |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This definition makes the assumption that there are no sites before subsystem A (that is, c = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We discuss in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' J why this does not affect the average correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By setting f = 1, we furthermore use the fact that the sites after subsystem B do not play a role as a result of the sequential generation [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (55)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (ii) We compute I(A : B) with respect to |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (iii) We repeat steps (i) and (ii) 10 000 times to compute the average of I(A : B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (iv) We repeat steps (i) through (iii) for r ∈ {7, 9, 11, 13, 15}, plot the averages of I(A : B) against r, and fit the data to extract the average correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (v) To obtain Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 3, we repeat steps (i) through (iv) for different d and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 32 Appendix L: Transfer Matrices in Two Dimensions This appendix expands on Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We will state the definitions of the boundary tensors and prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From the main text, recall that we define V (i,j) = U (i,j) ⊗ U (i,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' By computing the k-fold twirl, we obtain the building block = � dU = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L1) With that, we have E|ψ⟩⟨ψ|⊗k = = , (L2) where, in the final step, we have cut permutation-valued (green) legs instead of bond (blues) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' S is the tensor stated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' S′ and S′′ reflect the different boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Instead of first stating the tensors with bond (blue) legs, let us immediately state those with permutation-valued (green) legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The tensors at the top boundary are given via = = (L3) = � σ∈Sk Wg � στ −1, dD2� (dD)#(σρ)D#(σθ−1), (L4) and those at the right boundary are given via = = (L5) = � σ∈Sk Wg � στ −1, dD2� (dD)#(σρ)D#(σν−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L6) 33 We will always contract S′′ with P (d) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The tensor in the top-right corner thus plays the same role as the final vector |Fk⟩ = e1 ∈ Rk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' does in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In fact, Te = |Fk⟩⟨Fk|, = = ≡ (L7) = � σ∈Sk Wg � στ −1, dD2�� dD2�#(σe) = δeτ, (L8) where we have defined = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L9) If a tensor corresponding to e ∈ Sk is contracted with |Fk⟩, it factorizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' For tensors at the top boundary, we have = = = (L10) = � σ∈Sk Wg � στ −1, dD2�� dD2�#(σe) = δeτ, (L11) for those at the right boundary, we have = = = (L12) = � σ∈Sk Wg � στ −1, dD2�� dD2�#(σe) = δeτ, (L13) and for those in the bulk, we have = = = (L14) = � σ∈Sk Wg � στ −1, dD2�� dD2�#(σe) = δeτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L15) 34 The identities above lead to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (99): = = = (L16) = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (L17) Appendix M: Spectrum of the Transfer Matrix Te with e ∈ S2 In this appendix, we state and prove two lemmas concerning the spectrum of Te with e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let us start with some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define |0⟩ = � 1 0 � , |1⟩ = � 0 1 � , and |+⟩ = � 1 1 � , (M1) and map the contraction of tensors defining Te with e ∈ S2 to a multiplication of matrices: = = , (M2) where = � � � 1 α α γ 0 0 0 0 0 0 0 0 0 β β δ � � � (M3) with α = d2D3 − D d2D4 − 1 , β = dD3 − dD d2D4 − 1 , γ = d2D2 − D2 d2D4 − 1 , and δ = dD2 − d d2D4 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M4) Note that = 0 if i ̸= j, (M5) 35 and N = = � 1 α 0 β � (M6) is equal to Te with e ∈ S2 for d → dD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With that, it easy to check that = = ⟨o|N|i⟩⟨i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M7) We also introduce an analytical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We define Mj = I⊗(j−1) ⊗ M ⊗ I⊗(h−j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M8) Te with e ∈ S2 is then given by Te = � I⊗h ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗h� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M9) With i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , ih ∈ {0, 1} and o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , oh ∈ {0, 1}, the entries of Te with e ∈ S2 are given by � ⟨o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , oh| ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ |i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' , ih⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M10) Our first lemma states that Te with e ∈ S2 is block triangular, where our definition of blocks arises from the indexing of rows and columns in base 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, with 2 ≤ j ≤ h, the jth diagonal block of Te, which we denote by T (j) e ∈ R2j−1×2j−1, has fixed indices ih = · · · ij+1 = oh = · · · = oj+1 = 0 and ij = oj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M14), it is given by T (j) e = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M11) The first diagonal block, which we denote by T (1) e ∈ R2×2, is given by T (1) e = = = � 1 α 0 β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M12) In particular, we will prove that a block of Te is zero if its defining row digit oj is higher than its defining column digit ij, which implies that Te is upper block triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As the proof relies exclusively on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M7), Te inherits its upper block triangularity from the upper triangularity of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ S2 is upper block triangular because � I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ ⊗ |0⟩⊗(h−j)� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M13) 36 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From = |0⟩ (M14) and = 0, (M15) it follows that = = 0, (M16) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In our second lemma, we utilize the block triangularity of Te with e ∈ S2 to make a direct statement about its two leading eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let |λ1| > |λ2| > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, for any h, λ1 = 1 and λ2 = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Furthermore, λ1 and λ2 are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The spectrum of Te with e ∈ S2 is given by the union of the spectra of its diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is evident that the first block T (1) e has eigenvalues 1 and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the following, we show that any other diagonal block T (j) e with 2 ≤ j ≤ h can be written as a product of β and a strictly substochastic matrix, implying that its eigenvalues are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We structure the proof in steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show that any diagonal block T (j) e with 2 ≤ j ≤ h can be written as β times a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From = β|1⟩, (M17) it follows that T (j) e = = β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M18) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We argue that the matrix (M19) 37 is strictly column substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that = � � � 1 α α γ 0 0 0 0 0 0 0 0 0 β β δ � � � (M20) is column substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While the first column of M evidently sums to 1, the sums of the other columns are strictly bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As a result, Mj−1 · · · M1 is column substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The boundary condition |1⟩ does not affect this because it specifies a subset of columns of the matrix Mj−1 · · · M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In fact, it imposes strict substochasticity because this subset does not include the only column summing to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Also the boundary condition ⟨+| does not affect the substochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The boundary condition means that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M19) is a sum of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each of these matrices comprises a disjoint subset of rows of the matrix Mj−1 · · · M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because Mj−1 · · · M1, the matrix comprising the whole set of rows, is column substochastic, so is the sum of the matrices comprising the disjoint subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 and β are the only eigenvalues of T (1) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M20) is strictly column substochastic, the eigenvalues of any diagonal block T (j) e with 2 ≤ j ≤ h are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As a preparation for App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' N, we provide the proofs of Lemmas 6 and 7 in analytical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Lemma 6 in analytical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From � ⟨0| ⊗ ⟨0| � M � I ⊗ |0⟩ � = |0⟩ (M21) and � ⟨0| ⊗ ⟨0| � M � I ⊗ |1⟩ � = 0, (M22) it follows that � I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ ⊗ |0⟩⊗(h−j)� = � I⊗(j−1) ⊗ ⟨0| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ � (M23) = 0, (M24) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof of Lemma 7 in analytical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in the version with graphical notation, we structure the proof in steps, without repeating the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From � ⟨1| ⊗ ⟨0| � M � I ⊗ |0⟩ � = β|1⟩, (M25) it follows that T (j) e = � I⊗(j−1) ⊗ ⟨1| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |1⟩ � (M26) = β � I⊗(j−1) ⊗ ⟨1| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M27) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that � I⊗(j−1) ⊗ ⟨1| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� (M28) is strictly column substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 and β are the only eigenvalues of T (1) e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M28) is strictly column substochastic, the eigenvalues of any diagonal block T (j) e with 2 ≤ j ≤ h are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 38 Appendix N: Spectrum of the Transfer Matrix Te with e ∈ S4 In this appendix, we state and prove two lemmas concerning the spectrum of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Using the same notation as in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M, we will draw on results from that appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ S4 is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (M2), now with M ∈ R576×576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' it holds that � ⟨o| ⊗ ⟨0| � M � I ⊗ |i⟩ � = ⟨o|N|i⟩⟨i|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(N1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='N = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='⟨+| ⊗ I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='I ⊗ |0⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='1 α α α α α α γ γ γ γ γ γ γ γ η η η η η η ρ ρ ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 β 0 0 0 0 0 δ δ δ δ 0 0 0 0 θ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ σ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 β 0 0 0 0 δ δ 0 0 δ δ 0 0 ι θ θ θ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='σ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 β 0 0 0 0 0 δ δ δ δ 0 0 θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ θ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 β 0 0 δ δ 0 0 0 0 δ δ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ θ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 β 0 0 0 δ δ 0 0 δ δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ θ θ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='σ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 β 0 0 0 0 δ δ δ δ θ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='ι θ σ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 ε ζ 0 0 0 0 0 0 κ κ λ λ κ λ υ υ υ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 ζ ε 0 0 0 0 0 0 λ λ κ κ λ κ υ υ υ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 ε ζ 0 0 0 0 κ κ κ λ λ λ υ υ υ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 ζ ε 0 0 0 0 λ λ λ κ κ κ υ υ υ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 ε ζ 0 0 κ λ κ κ λ λ υ υ υ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 ζ ε 0 0 λ κ λ λ κ κ υ υ υ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 0 0 ε ζ κ λ λ κ κ λ υ υ υ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 0 0 ζ ε λ κ κ λ λ κ υ υ υ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 µ ν ν ν ν ξ τ ϕ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ν µ ν ξ ν ν τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ν ν µ ν ξ ν ϕ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ν ξ ν µ ν ν τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ν ν ξ ν µ ν ϕ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ξ ν ν ν ν µ τ ϕ τ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='(N2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content='is equal to Te with e ∈ S4 for d → dD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M, our first lemma states that Te with e ∈ S4 is block triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The definition of blocks now arises from the indexing of rows and columns in base 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, any diagonal block (but the first) is defined by ih = · · · = ij+1 = oh = · · · = oj+1 = 0 and ij = oj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' There are four classes of diagonal blocks: The first diagonal block is in its own class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is given by � I ⊗ ⟨0|⊗(h−1) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I ⊗ |0⟩⊗(h−1)� = � I ⊗ ⟨0| � M � |+⟩ ⊗ I � = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N3) The second class of diagonal blocks corresponds to transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ij and oj correspond to the same transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' There are six subblocks in this class because there are six different transpositions in S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The third class of diagonal blocks corresponds to permutations with a single fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ij and oj correspond to any of the two permutations with the same single fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' There are four subblocks in this class because there are four different choices of a single fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The fourth class of diagonal blocks corresponds to permutations with no fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ij and oj correspond to any of the nine permutations with no fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, we will prove that a block of Te is zero if its defining row digit oj is higher than its defining column digit ij stand in a certain relation to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As the proof relies exclusively on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N1), Te again inherits its upper block triangularity from the upper triangularity of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Te with e ∈ S4 is upper block triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 39 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N1), it follows that � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� = 0 (N4) if ij corresponds to the trivial permutation and oj does not, ij corresponds to a transposition and oj corresponds to a different transposition or a permutation with one or no fixed point, ij corresponds to a permutation with a single fixed point and oj corresponds to a permutation with a different single fixed point or no fixed point, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In our second lemma, we utilize the block triangularity of Te with e ∈ S4 to make direct a statement about its two leading eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Let |λ1| > |λ2| > · · · ≥ 0 denote the distinct eigenvalues of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, for any h, λ1 = 1 and λ2 = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Furthermore, λ1 is non-degenerate, and λ2 has a degeneracy of six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As is the case for Te with e ∈ S2, the spectrum of Te with e ∈ S4 is given by the union of the spectra of its diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The two leading eigenvalues of the first diagonal block N are 1 and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In the following, we show that any other diagonal block can be written as a product of β and a matrix whose spectral radius is strictly bounded by 1, implying that its eigenvalues are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We again structure the proof in steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We show that any diagonal block but the first can be written as a product of beta and a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' it follows that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' if ij and oj correspond to the same transposition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� = � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ � (N5) = β � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N6) if ij and oj correspond to any two permutations with the same single fixed point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� = � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ � (N7) = p � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� (N8) < β 2 � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N9) where {ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ζ} ∋ p < β/2 [63] depends on ij and oj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' if ij and oj correspond to any permutation with no fixed point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0|⊗(h−j) ⊗ ⟨0| �� Mh · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ ⊗ |0⟩⊗(h−j)� = � I⊗(j−1) ⊗ ⟨oj| ⊗ ⟨0| �� Mj · · · M1 �� |+⟩ ⊗ I⊗(j−1) ⊗ |ij⟩ � (N10) = p � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� (N11) < β 9 � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N12) where {µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' χ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' ψ} ∋ p < β/9 [63] depends on ij and oj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 40 Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We now argue that the spectral radius of the matrix � I⊗(j−1) ⊗ ⟨oj| �� Mj−1 · · · M1 �� |+⟩ ⊗ I⊗(j−1)� (N13) is strictly bounded by 1 for 2 ≤ j ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It holds that the spectral radius of M is bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' While the first column of |M| sums to 1, the sums of the other columns are strictly bounded by 1 [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As a result, the spectral radius of Mj−1 · · · M1 is bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The boundary condition |o⟩ does not affect this because it specifies a subset of columns of the matrix Mj−1 · · · M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In fact, it imposes a strict bound because this subset does not include the only column summing to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Also the boundary condition ⟨+| does not affect the bound on the spectral radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The boundary condition means that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N13) is a sum of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Each of these matrices comprises a disjoint subset of rows of the matrix Mj−1 · · · M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because the spectral radius of Mj−1 · · · M1, the matrix comprising the whole set of rows, is bounded by 1, so is the sum of the matrices comprising the disjoint subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 1 and β are the two leading eigenvalues of the first diagonal block N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' They are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because the spectral radius of the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (N13) is strictly bounded by 1, the eigenvalues of any other diagonal block are strictly smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix O: Proof of Result 4 Result 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of I2(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps, following the structure of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The steps are overall very similar to those of that proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EI2(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, we make the assumption that E log(X) = log(EX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Then, EI2(A : B) = log � E tr � ϱ2 AB �� − log � E tr � ϱ2 A �� − log � Etr � ϱ2 B �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (O1) E tr � ϱ2 A � , E tr � ϱ2 B � , and E tr � ϱ2 AB � can be written in the desired form [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (62) and (63)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EI2(A : B) in terms of the transfer tensors defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V A and use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (100) to map contractions of two-dimensional tensor networks to multiplications of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The latter is enabled by our definition of subsystems A and B [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We have done this for E tr � ϱ2 A � in graphical notation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V C [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (104) and (105)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is easy to confirm that EI2(A : B) = log � ⟨I2|T c e T a (12)T r e T b (12)|F2⟩ � − log � ⟨I2|T c e T a (12)|F2⟩ � − log � ⟨I2|T c+a+r e T b (12)|F2⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (O2) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EI2(A : B) in terms of the spectrum of Te with e ∈ S2, which we consider in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we know λ1 and λ2 as well as their algebraic and geometric multiplicities, we do not need Te to be diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Expanding T c e and taking the limit c → ∞ yields EI2(A : B) = log � ⟨L1|T a (12)T r e T b (12)|F2⟩ � − log � ⟨L1|T a (12)|F2⟩ � − log � ⟨L1|T b (12)|F2⟩ � , (O3) where we have used that ⟨I2|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding T r e and using that |F2⟩ = |R1⟩, we have I2(A : B) = log � ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + λr 2⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ + O � rv−1λr 3 �� − log � ⟨L1|T a (12)|R1⟩ � − log � ⟨L1|T b (12)|R1⟩ � , (O4) where v denote the size of the largest Jordan block with respect to λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 41 Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EI2(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' With Λ = max �� λ2r 2 , rv−1λr 3 �� , EI2(A : B) = log � 1 + λr 2 ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + O � rv−1λr 3 � � (O5) = λr 2 ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ + O(Λ) (O6) ≡ K exp � −r ξ � + O � exp � − r χ �� , (O7) where K = ⟨L1|T a (12)|R2⟩⟨L2|T b (12)|R1⟩ ⟨L1|T a (12)|R1⟩⟨L1|T b (12)|R1⟩ (O8) and ξ = − 1 log(λ2) = − � log �dD3 − dD d2D4 − 1 ��−1 = ξ2D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (O9) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Appendix P: Proof of Result 5 Result 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The average of N(A : B) with respect to the random isoTNS ensemble and subsystems A and B as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a) decays exponentially as specified in Definition 1 with the average correlation length ξ2D defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We split the proof into four steps, following the structure of the proof of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The steps are overall very similar to those of the proof of Result 2 (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We rewrite EN(A : B) in terms of expressions of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in one dimension, with the Hilbert-Schmidt inner product, EN(A : B) = E tr � ϱ2 AB � + E tr � ϱ2 A � tr � ϱ2 B � − 2E tr[ϱAB(ϱA ⊗ ϱB)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P1) The right-hand side can be written in the desired form [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We express EN(A : B) in terms of the transfer tensors defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' V A and use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (100) to map contractions of two-dimensional tensor networks to multiplications of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The latter is enabled by our definition of subsystems A and B [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' 4 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' It is easy to confirm that EN(A : B) = ⟨I4|T c e T a (12)T r e T b (12)|F4⟩ + ⟨I4|T c e T a (34)T r e T b (12)|F4⟩ − 2⟨I4|T c e T a (12)T r e T b (13)|F4⟩ (P2) = ⟨I4|T c e T a (12)T r e � T b (12) + T b (34) − 2T b (13) � |F4⟩ (P3) ≡ ⟨I4|T c e AT r e B|F4⟩, (P4) where we have defined A = T a (12) and B = T b (12) + T b (34) − 2T b (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P5) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We expand EN(A : B) in terms of the spectrum of Te with e ∈ S4, which we consider in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Because we know λ1 and λ2 as well as their algebraic and geometric multiplicities, we do not need Te to be diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Expanding T c e and taking the limit c → ∞ yields EN(A : B) = ⟨L1|AT r e B|F4⟩, (P6) 42 where we have used that ⟨I4|R1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' After expanding T r e and using that |F4⟩ = |R1⟩, we have EN(A : B) = ⟨L1|A|R1⟩⟨L1|B|R1⟩ + λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3) (P7) = λr 2 w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ + O(λr 3), (P8) where, in the second line, we have used that ⟨L1|B|R1⟩ = 0, (P9) which we prove in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' As in the proof of Result 2 (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' H), we prove that ⟨L1|T b t |R1⟩ does not depend on the two elements the transposition t ∈ S4 acts upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In fact, the proof follows from the proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (H12) of that appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' We just need two additional considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' First, the proof of Lemma 8 is not specific to the trivial permutation e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' In particular, Tρ exhibits an upper block triangular structure for any ρ ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' The first diagonal block is given by Tρ with ρ ∈ S4, � ⟨si|Tρ|sj⟩ � 1≤i≤24 1≤j≤24 = Tρ, (P10) which implies that � ⟨si|T b ρ |sj⟩ � 1≤i≤24 1≤j≤24 = T b ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P11) Second, the eigenvectors ⟨L1| and |R1⟩ of Te with e ∈ S4 arise from those of Te with e ∈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, |R⟩1 = s1 and � ⟨L1|si⟩ � 1≤i≤7 = � 1 α β − 1 α β − 1 α β − 1 α β − 1 α β − 1 α β − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P12) ⟨L1|T b t |R1⟩ thus does not depend on the two elements the transposition t ∈ S4 acts upon because ⟨L1|T b t |R1⟩ does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' This implies that ⟨L1|B|R1⟩ = ⟨L1| � T b (12) + T b (34) − 2T b (13) � |R1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P13) Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' Finally, we can write EN(A : B) in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' That is, EN(A : B) ≡ K exp � −r ξ � + O � exp � − r χ �� , (P14) where K = w2 � µ=1 ⟨L1|A|R(µ) 2 ⟩⟨L(µ) 2 |B|R1⟩ (P15) and ξ = − 1 log(λ2) = − � log �dD3 − dD d2D4 − 1 ��−1 = ξ2D > χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'} +page_content=' (P16) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE3T4oBgHgl3EQfnQrW/content/2301.04624v1.pdf'}