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+Revisiting Pure State Transformations with Zero Communication
+Ian George and Eric Chitambar
+Electrical and Computer Engineering Department, University of Illinois at Urbana-Champaign
+(Dated: January 13, 2023)
+It is known that general convertibility of bipartite entangled states is not possible to arbitrary error
+without some classical communication. While some trade-offs between communication cost and con-
+version error have been proven, these bounds can be very loose. In particular, there are many cases
+in which tolerable error might be achievable using zero-communication protocols. In this work we
+address these cases by deriving the optimal fidelity of pure state conversions under local unitaries as
+well as local operations and shared randomness (LOSR). We also use these results to explore catalytic
+conversions between pure states using zero communication.
+I.
+INTRODUCTION
+The theory of quantum mechanics through the
+lens of information and vice versa [1–3] has af-
+forded the physicist and the information scientist
+alike with a new way to view the objects and long-
+term goals of their study.
+No better example of
+this can be found than quantum resource theo-
+ries. Quantum resource theories specify the rele-
+vant physical property in such a manner as to better
+tease apart the complexities of quantum mechanics
+while also establishing what tasks may be achieved
+with said resource [4]. Perhaps the earliest example
+of such a resource theory is the resource theory of
+entanglement. Entanglement may be viewed as a
+form of correlation that does not exist in the classi-
+cal world [5]. Roughly speaking, the resource the-
+ory of entanglement asks (1) what tasks may be per-
+formed better using entangled states and (2) how
+entangled states may be converted from one to an-
+other under some class of free operations.
+The most standard view of the resource theory
+of entanglement considers the set of free operations
+to be local operations and classical communication
+(LOCC) which captures the ‘distant lab’ paradigm
+where two (or more) parties share an entangled
+state in spatially separated labs and they can only
+perform operations on their respective portions and
+exchange classical information (See Fig.
+1).
+Not
+only is this the most standard set of free operations,
+but in some respect it seems minimal. Indeed, Hay-
+den and Winter showed that to convert one (pure)
+entangled state to another to sufficiently small pre-
+cision requires a certain amount of communication
+between labs, regardless of how many auxiliary
+EPR pairs they share [6] (see also [7]). This is dis-
+tinct not only from the classical setting [8], but also
+from quantum states that are not entangled [9, 10].
+However, the results of Hayden and Winter, while
+fundamental, do not give us a complete picture of
+the tradeoff between communication and achiev-
+able tolerated error in pure state conversions. In-
+(a)
+(b)
+FIG. 1: Conversion of pure states in distant labs. (a) The
+LOCC model where communication is exchanged. (b)
+The embezzling of quantum states where an auxiliary
+entangled state is used. This may be seen as a special
+case of catalytic conversion.
+deed, it is easy to find examples of state conversions
+which, according to the best known lower bounds,
+still may be possible to perform with a tolerated er-
+ror of 1% using no communication (see Example 1
+of Section III). This show that a relatively large gap
+in our understanding of zero-communication en-
+tanglement transformations still persists, and one
+we aim to address in this work.
+Moreover, the tools we develop to address this
+problem will also allow us to study pure state trans-
+formations using shared auxiliary entanglement.
+The operational paradigm in which parties are al-
+lowed to use arbitrary pre-shared entanglement but
+no communication is known as local operations and
+shared entanglement (LOSE) [11].
+By itself, the
+problem of pure state convertibility |ψ⟩AB → |φ⟩AB
+under LOSE is trivial since Alice and Bob could al-
+ways just demand |φ⟩ as their pre-shared entangle-
+ment and then throw away |ψ⟩ when it is given.
+However, if one demands that the pre-shared en-
+tanglement is also returned in addition to the target
+state |φ⟩, then the problem becomes quite interest-
+ing, i.e. |ψ⟩AB ⊗ |ω⟩A′B′ → |φ⟩AB ⊗ |ω⟩A′B′ for aux-
+iliary pre-shared entanglement |ω⟩A′B′. Transfor-
+mations of this form are known as catalytic trans-
+formations with |ω⟩A′B being the catalyst. Remark-
+ably, van Dam and Hayden have shown that there
+exists a family of entangled catalysts, known as
+arXiv:2301.04735v1  [quant-ph]  11 Jan 2023
+
+A
+BA
+A
+A'
+A'
+B'
+B'
+B
+B2
+FIG. 2: Comparison of [12] (dark pink),[17] (dark
+green), and this work’s results (blue). [17] finds lower
+bounds on the classical communication necessary to
+convert one state to another, but in the zero
+communication setting these are too loose. We find
+methods for solving this exactly (Section IV), which
+establishes that communication is necessary for larger
+tolerated errors. [12] establishes a method for pure state
+transformations with zero communication with massive
+amounts of entanglement, but it scales inversely with
+the error, which we find can be too strong for a relevant
+error range, even if ultimately it is optimal (Section VI).
+universal embezzling states [12], such that for any
+tolerated non-zero error one can always prepare a
+pure state using a member of this family and zero
+communication. More amazingly, they showed that
+as the error tends to zero, it is roughly optimal since
+it scales nearly the same as if you add LOCC and
+allow the catalyst to be state dependent. This near
+optimality along with Hayden and Winter’s result
+has, understandably, largely ceased the study of en-
+tanglement transformations with zero communica-
+tion, because when one needs entanglement trans-
+formations without communication, one uses em-
+bezzlement [13, 14].1 It is however not clear what
+is the necessary error for embezzlement to become
+near optimal, which could be relevant in practical
+settings. Indeed, for any tolerated error, it is easy to
+find sufficient conditions on pure states to be con-
+verted with no catalyst at all (Example 2 of Section
+III). This is an indication that we also do not under-
+stand embezzling and catalytic convertibility suffi-
+ciently well.
+1 The notable exceptions to this halted topic of research has been
+the consideration of special embezzling families [15] and the
+correlated sampling lemma [16], which may be viewed as a
+variation of embezzling.
+A.
+Summary of Results
+The primary aim of this work is to provide tighter
+lower bounds on the error in pure state entangle-
+ment convertibility with zero communication.
+A
+high level comparison of our results to the afore-
+mentioned work on this topic are presented in Fig.
+2. This depicts a ‘one-shot resource tradeoff’ region
+that must contain the ‘true’ one-shot resource trade-
+off surface for a given pure state conversion. Hay-
+den and Winter’s result provides a lower bound
+on the achievability independent of the amount of
+shared maximally entangled states, but their result
+can be too loose when considering zero communi-
+cation. van Dam and Hayden’s result provides an
+outer bound on the achievability surface on the face
+pertaining to LOSE, but their result in fact can be
+too loose when the error is not sufficiently small.
+In this work, our results allow one to exactly solve
+the minimal error in the zero communication set-
+ting and also provide significantly tighter bounds
+than quantum embezzling for a relevant region on
+the LOSE face (See Fig. 2).
+To formally establish our results, we reduce the
+class of questions regarding optimal pure state con-
+version to optimization problems that only concern
+probability distributions. This is because of a bijec-
+tion between the equivalence classes of pure states
+under local unitaries— which are defined solely
+by their Schmidt coefficients— and the probability
+simplex. We do this by showing the optimal fidelity
+of pure state transformations with local unitaries
+is efficiently computable.
+Of course, in general
+one would not expect local unitaries to be the op-
+timal strategy and we build on this result to present
+a non-convex optimization program over an opti-
+mization variable with bounded dimension. An im-
+mediate corollary of this result is the impossibility
+of pure state conversions with zero communication
+for negligible error. We also present efficient com-
+putable upper bounds on the achievable error using
+a semidefinite programming (SDP) relaxation. We
+also show that in the case where either the seed (i.e.
+initial) or target state is a two-qubit state, the local
+unitary strategy is optimal. However, we can show
+for larger dimensions this is not the case.
+Having established general properties in the sin-
+gle copy case, we move to the multiple copy case,
+i.e. where the seed and/or target state is of inde-
+pendent and identically distributed (i.i.d.)
+form.
+This is standard in determining the rate of con-
+verting one state to another. In particular, we con-
+sider dilution and distillation where the seed state
+or target state respectively is many copies of a max-
+imally entangled state and show these are convex
+optimization programs and may be seen as involv-
+
+Entanglement
+LOSE
+1
+Tolerated
+0
+Error ε
+Gap
+LOCC
+Classical
+Communication3
+ing the Ky-Fan norms when extended to the regime
+where they are not a norm. Lastly, in a sense ex-
+tending our earlier two-qubit results, we establish
+that if the target state is an n−fold copy of a two-
+qubit entangled state and the seed state’s Schmidt
+rank is less than the target state, then local unitaries
+are the optimal strategy.
+Finally, given these results, we turn our atten-
+tion to quantum embezzlement. We begin by not-
+ing that the correspondence between Schmidt co-
+efficients and probability distributions means that
+quantum embezzlement implies a classical equiv-
+alent we call randomness embezzlement. We then
+proceed to use our new tools to consider the prob-
+lem of catalyzed pure state conversion under local
+unitaries, in effect a generalization of embezzling,
+and compare it to embezzling.
+We show in par-
+ticular that at least in general the optimality of the
+embezzling states is only for very small errors. In-
+deed, we show for reasonable tolerable errors, the
+embezzling state may have a Schmidt rank of many
+orders of magnitude larger than a simple catalyst.
+This may have practical relevance and strongly re-
+fines our understanding of pure state transforma-
+tions under LOSE.
+Organization of the Paper
+The rest of the paper
+is organized as follows. In Sections II and III we
+present the necessary notation and background re-
+spectively to understand the rest of the paper. In
+Section IV, we
+• Make explicit the correspondence between
+pure states under LU and the probability sim-
+plex and note this implies the existence of a
+classical variation of embezzlement (Theorem
+2)
+• Prove our equation for fidelity of state con-
+version under local unitaries (Theorem 5) and
+our optimization for fidelity of state conver-
+sion under local operations and shared ran-
+domness (Theorem 6)
+• Establish computable upper bounds on the fi-
+delity of state conversion under LOSR (Theo-
+rem 9).
+In Section V we present the results where the tar-
+get or seed state is of i.i.d. form. In Section VI we
+discuss catalysts under local unitaries, the general
+frameworks that includes quantum embezzlement,.
+In Section VII we discuss why our theory does not
+generalize beyond bipartite pure states.
+II.
+NOTATION
+Our notation largely aligns with standard texts
+[18, 19].
+In this paper we consider finite dimen-
+sional quantum systems. Given n ∈ N, we define
+[n] := {1, ..., n}. A finite dimensional Hilbert space
+will be labeled with a capital roman letter, e.g. A, B,
+etc. As they are finite dimensional, these Hilbert
+spaces may be identified by the isomorphism A ∼=
+Cd where d ∈ N. The space of linear maps from a
+Hilbert space A into itself, i.e. the space of endo-
+morphisms, is denoted L(A). The space of quan-
+tum states, or density matrices, with respect to a
+Hilbert space A, is the space of positive semidefi-
+nite operators with unit trace, i.e. D(A) := {ρ ∈
+L(A) : ρ ⪰ 0 & Tr(ρ) = 1} where ⪰ is the L¨owner
+order. If a quantum state is a joint state over multi-
+ple Hilbert spaces, we will use a subscript to specify
+this, e.g. ρAB ∈ D(A ⊗ B). We say a quantum state
+ρA ∈ D(A) is pure if Tr
+�
+ρ2
+A
+� = 1 which is equiva-
+lent to there being a unit vector |ψ⟩ ∈ A such that
+ρA = |ψ⟩⟨ψ|, where we are using bra-ket notation.
+For this previous reason, we generally just specify
+a pure state by |ψ⟩A, or ψ if we are considering its
+density matrix representation. We denote the space
+of pure states S(A), where S stands for unit sphere.
+A state is classical if it is diagonal in a specific
+choice of basis for L(A). We call this the computa-
+tional basis. The space of classical probability dis-
+tributions over d elements, the probability simplex
+which we denote P(d), may be viewed as the set
+of non-negative d−dimensional vectors that sum to
+one or the set of diagonal density matrices in the
+computational basis.
+To distinguish between the
+two, we write P for the matrix version and p for
+the vector version. We also define the set of entry-
+wise decreasing probability distributions over d el-
+ements, i.e. elements of the form p↓(1) ≥ p↓(2) ≥
+... ≥ p↓(d), by P↓(d).
+A quantum channel E
+∈
+C(A, B) is a (lin-
+ear) completely positive, trace preserving map E :
+L(A) → L(B). Any quantum channel admits an
+isometric representation, e.g. given E ∈ C(A, B),
+there exists a Hilbert space E such that |E| ≤ |A||B|
+and isometry V : A → B ⊗ E such that Φ(X) =
+TrE(VXV†) where TrE is the partial trace on the E
+space and X† is the Hermitian conjugate.
+Given the space of linear operators from A ∼= Cd
+to B ∼= Cd′, L(A, B), the vec mapping vec : L(A ⊗
+B) → A ⊗ B is defined by vec(|i⟩ ⟨j|) = |j⟩ ⊗ |i⟩
+where {|i⟩}i∈[d] and {|j⟩}j∈[d′] are the computa-
+tional bases for A and B respectively. This choice
+of vec mapping satisfies the identity
+(XT
+1 ⊗ X0) vec(Y) = vec(X0YX1) ,
+(1)
+where X0 ∈ L(A0, B0), X1 ∈ L(A1, B1), and Y ∈
+L(B1, B0). The vec mapping is also an isometry in
+the sense that for all X, Y ∈ L(A, B),
+⟨X, Y⟩ = ⟨vec(X), vec(Y)⟩ ,
+
+4
+where ⟨·, ·⟩ on the L.H.S. is the inner product on
+L(A, B) defined by ⟨X, Y⟩
+=
+Tr
+�
+X†Y
+�
+and the
+R.H.S. is the inner product on vectors A ⊗ B defined
+by ⟨ψ|φ⟩ = ∑i ψ(i)φ(i) where · is the conjugate.
+III.
+BACKGROUND & MOTIVATION
+Throughout this section we fix A ∼= Cd, B ∼= Cd′
+for clarity.
+a.
+Fidelity
+The fidelity is a standard measure of
+similarity between two positive semidefinite oper-
+ators R, S ≥ 0.
+F(R, S) =
+���
+√
+R
+√
+S
+���
+2
+1 = Tr
+��√
+SR
+√
+S
+�2
+,
+(2)
+where the square root of a positive semidefinite
+operator is defined in the standard fashion on its
+spectral decomposition and ∥ · ∥1 is the Schatten
+1−norm. It satisfies various properties that will be
+relevant for this work which we summarize here.
+All of these may be verified by direct calculation or
+by referring to standard texts.
+Proposition 1 (Summary of Fidelity Properties).
+Let ρ, σ ∈ D(A). The following hold:
+1. 0 ≤ F(ρ, σ) ≤ 1 where the upper bound is
+saturated if and only if ρ = σ and the lower
+bound saturates if and only if their images are
+orthogonal.
+2. The fidelity is isometrically invariant, i.e.
+given isometry V : A → B,
+F(VρV†, VσV†) = F(ρ, σ) .
+3. The fidelity satisfies data-processing. That is,
+for any quantum channel E ∈ C(A, B),
+F(ρ, σ) ≤ F(E(ρ), E(σ)) .
+4. If both states are pure,
+F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = | ⟨ψ|φ⟩ |2 ,
+and if one state is pure
+F(|φ⟩⟨φ| , σ) = ⟨φ| σ |φ⟩ .
+5. If both states are classical, P, Q ∈ P(d), then
+the fidelity reduces to the square of the Bhat-
+tacharyya coefficient:
+F(P, Q) =
+�
+� ∑
+i∈[d]
+�
+p(i)q(i)
+�
+�
+2
+= BC(p, q)2 ,
+where p(i) = P(i, i) and likewise for Q.
+6. Given pure states with the same eigenbasis
+and real amplitudes, |ψ⟩ = ∑x
+�
+p(x) |x⟩,
+|φ⟩ = ∑x
+�
+q(x) |x⟩ , the fidelity reduces to
+the square of the Bhattacharyya coefficient of
+the probability distributions defined by the
+amplitudes:
+F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2 .
+We also note that in all of these definitions there
+is a pesky squaring that effectively we don’t care
+about. For this reason we could define the square
+root fidelity:
+√
+F(R, S) :=
+�
+F(R, S) .
+Note the square root fidelity could be viewed as
+the quantum extension of the Bhattacharyya coef-
+ficient.
+b.
+Norms
+In defining the fidelity we used the
+Schatten 1−norm.
+More generally, there are the
+Schatten p−norms which for X ∈ L(A, B) may be
+defined as ∥X∥p := ∥σ(X)∥p where σ(X) is the
+ordered vector of singular values of X, σ1(X) ≥
+σ2(X) ≥ ... ≥ σrank(X)(X) and it is being evalu-
+ated under the Lp−norm where p ≥ 1.
+The in-
+finity norm, ∞−norm, is limp→∞ ∥X∥p = ∥X∥∞ =
+maxi σi(X). The infinity norm was generalized to
+the Ky Fan k−norms ∥X∥(k) := ∑ σi(X) for 1 ≤ k ≤
+min{d, d′}. The Ky Fan norms have relevance in
+measuring entanglement [20]. A generalization of
+the Ky Fan and Schatten norms together is given by
+the (k, p)−norms [21]
+∥X∥(k,p) :=
+�
+� ∑
+i∈[k]
+σi(X)p
+�
+�
+1/p
+,
+(3)
+which also have use in measuring entanglement
+of pure states [22].
+Much like is common to
+do for the Schatten p−norms, we can extend the
+(k, p)−norms to p > 0 with the caveat they won’t
+be norms as they won’t in general satisfy subaddi-
+tivity (the triangle inequality) for p ∈ [0, 1).
+c.
+Entanglement Theory
+A bipartite quantum
+state ρAB is separable if there exists n ∈ N, p ∈
+P(n), {σi
+A}i∈[n] ⊂ D(A), and {τi
+B}i∈[n] such that
+ρAB = ∑
+i∈[n]
+p(i)σi
+A ⊗ τi
+B .
+Otherwise the state is entangled. As a pure state
+|ψ⟩⟨ψ|AB is defined by a unit vector, this reduces to
+a pure state is separable, referred to product in this
+setting, if and only if there exists |φ⟩A , |ϕ⟩B such
+
+5
+that |ψ⟩ = |φ⟩A ⊗ |ϕ⟩B. While this is sufficient for
+determining if a bipartite pure state is entangled,
+there is also a notion of ‘how’ entangled a state is
+in terms of Schmidt rank. Every bipartite pure state
+|ψ⟩AB admits a unique (up to re-ordering) decom-
+position of the form
+|ψ⟩AB = ∑
+i∈[k]
+�
+p(i) |ui⟩A ⊗ |vi⟩B ,
+(4)
+where
+k
+=
+max{d, d′},
+p
+∈
+P(k)
+and
+{|ui⟩}i∈[k], {|vi⟩}i∈[k] are orthonormal bases of A
+and B respectively. The
+�
+p(i) > 0 terms are re-
+ferred to as the Schmidt coefficients. The Schmidt
+rank of |ψ⟩AB, SR(|ψ⟩) = supp(p), i.e. the num-
+ber of Schmidt coefficients.
+This may be viewed
+as a measure of entanglement in the sense that the
+Schmidt rank of a product state is 1 and the maxi-
+mally entangled state |Φ+⟩CdCd =
+1
+√
+d ∑i |i⟩Cd |i⟩Cd
+has Schmidt rank d. We define the set SR(d) :=
+{|ψ⟩ : SR(|ψ⟩) ≤ d}, where we note this set is in-
+dependent of the dimension the state is embedded
+in.
+Lastly we note a particularly nice property of
+pure states, known as Uhlmann’s theorem.
+Lemma 1 (Uhlmann’s Theorem). Given ρ, σ
+∈
+D(A) and |ψ⟩ ∈ A ⊗ B such that TrB(ψ) = ρ, then
+F(ρ, σ) = max{| ⟨ψ|φ⟩ |2 : |φ⟩ ∈ A ⊗ B , TrB(φ) = σ} .
+d.
+No-Go Theorems, Embezzling, & Motivation
+With the established background, we now present
+the previous results related to zero communication
+pure state transformations which we will discuss
+our results in relation to. The first is a lower bound
+on the number of qubits or classical bits necessary
+to convert between pure states [6].
+Proposition 2. ([6, Theorem 8]) Consider a state
+transformation via channel E ∈ C(A ⊗ B, A ⊗ B)
+from seed state |φ⟩AB to target state |ψ⟩AB such
+that F(E(φ), ψ) ≥ 1 − ε. Then, independent of any
+amount of entanglement assistance, for δ =
+8√ε, in
+the implementation of E, q qubits were exchanged
+where
+q ≥1
+2 [∆δ(TrB(|φ⟩⟨φ|)) − ∆0(TrB(|ψ⟩⟨ψ|))]
++ log(1 − δ) ,
+(5)
+where
+exp(∆ε(P)) = min rank( �P) · λmax( �P)
+s.t. Tr
+�
+�P
+�
+≥ 1 − ε
+�P = ΠPΠ
+[P, Π] = 0
+Π2 = Π .
+Moreover, the bound given in (5) holds for a neces-
+sary amount of classical communication by multi-
+plying the R.H.S. by two.
+While the above proposition is very powerful
+and implies two states with different Schmidt de-
+compositions cannot be perfectly converted with
+zero communication, it is not sufficient in every sce-
+nario. In particular, the following example shows
+that in certain cases Proposition 2 cannot eliminate
+any state from being able to be converted to a given
+target state with relatively high fidelities.
+Example 1 (On the necessity of communication).
+Up to local unitaries, let the target state be |ψ⟩ =
+0.54 |00⟩ + 0.02 |11⟩ + 0.44 |22⟩, the seed state be any
+state |φ⟩ = ∑i∈[k]
+�
+p(i) |vi⟩ |ui⟩, and assume we
+are interested in a state transformation E such that
+F(E(φ), ψ) = 0.99. Then ε = 0.01, so δ > 0.56.
+One may verify ∆δ(P) = log(|1| · 0.44) < −1.18,
+by removing the 0.02 and 0.54 eigenvalues of P. It
+may be shown [6] that ∆0(TrB(|ψ⟩⟨ψ|)) ≥ 0, and
+log(1 − δ) < 0. It follows that in this setting the
+R.H.S. of (5) is negative.
+Therefore, we have no
+proof from this bound that any transformation for
+any seed state which achieves this relatively high
+fidelity of 99% requires any communication.
+While the above example shows there are reason-
+ably small tolerated errors ε where Proposition 2 is
+not helpful, when the tolerated error is sufficiently
+small, it will imply the need for communication.
+This sort of structure for sufficiently small ε also
+appears when considering quantum embezzlement
+[12], which may be seen as a solution to Proposition
+2 implying communication is necessary. Quantum
+embezzlement in effect shows one can make pure
+state transformations with zero communication to
+any non-zero error if they have the right sufficiently
+large entangled catalyst.
+Proposition 3. ([12]) Consider the family of catalyst
+states |µ(n)⟩A′B′ =
+1
+√Hn ∑n
+j=1
+1√
+j |j⟩A′ |j⟩B′ where
+Hn := ∑n
+i=1 n−1 is the Harmonic number. For any
+ε > 0 and target bipartite pure state |ψ⟩AB with
+Schmidt rank m, for n > m1/ε there exist unitaries
+UAA′, WBB′ such that
+F(UAA′ ⊗ WBB′(|µ(n)⟩A′B′ |0⟩A |0⟩B),
+|µ(n)⟩A′B′ ⊗ |ψ⟩AB) ≥ 1 − ε .
+Moreover, U, W are in effect permutations on the
+joint Schmidt bases.
+One can see quantum embezzlement implies a
+way to convert one pure state to another to non-
+zero error by picking a large enough catalyst and
+
+6
+then first ‘embezzling out’ the original state (un-
+computing |φ⟩ to |0⟩ |0⟩ via embezzling) and then
+‘embezzling in’ the target state |ψ⟩.
+What is perhaps most remarkable about the
+above approach is that it was shown in the original
+work that even if we allow LOCC and a state de-
+pendent catalyst, the error scales as Ω(1/ log(n))
+whereas for the above result the errors scales as
+O(1/ log(n)), so as the error is driven down, em-
+bezzling is near optimal. That is, as ε → 0, this strat-
+egy is effectively optimal. However, just as with the
+discussion pertaining to Proposition 2, it’s clear em-
+bezzling isn’t necessary for reasonable error levels
+in general. In fact, we show in the following ex-
+ample that for any non-zero error there exist states
+which can be converted without any catalyst.
+Example 2 (On the necessity of embezzling). As
+noted, as ε → 0, embezzling is necessary. However,
+it is not in general clear at what point embezzling
+becomes necessary.
+This can be seen as follows.
+Consider ε ∈ (0, 1) and two probability distribu-
+tions p, q ∈ P(m) such that the BC(p, q)2 ≥ 1 − ε.
+Define the seed state as |φ⟩ = ∑i∈[m]
+�
+p(i) |i⟩A |i⟩B
+and the target state as |ψ⟩ = ∑i∈[m]
+�
+q(i) |i⟩A |i⟩B.
+Then we have
+F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2 ≥ 1 − ε ,
+where we have used Item 5 of Proposition 1. There-
+fore, given |φ⟩, it requires no communication or en-
+tanglement to generate |ψ⟩ to error ε. In fact, as we
+show later (Proposition 4), this will be true for con-
+verting the set of states with Schmidt coefficients
+defined via p to the set of states with Schmidt coef-
+ficients defined via q in general.
+Given these two examples, we see that while
+these results give strong characterizations of pure
+state transformations with zero communication,
+neither the need for communication by Proposition
+2 nor the optimality of Proposition 3 when the error
+tends to zero give us a full understanding of this
+setting. It would therefore be of value to better un-
+derstand this task, and this is what the rest of this
+work addresses.
+IV.
+SINGLE COPY PURE STATE CONVERSION
+WITH ZERO COMMUNICATION
+Our primary goal of this section is to deter-
+mine the minimal error of conversion between pure
+states with zero communication, which would re-
+solve the gap presented in Example 1. To do this,
+we will use the correspondence between the prob-
+ability simplex and Schmidt coefficients under lo-
+cal unitaries (LU), which we establish in the follow-
+ing subsection. We also note that this implies the
+existence of a classical equivalent of embezzling,
+which we call randomness embezzling (Theorem
+2). This correspondence motivates the idea that the
+optimal fidelity of pure state conversion under local
+unitaries is simply re-ordering the Schmidt coeffi-
+cients, which we in fact prove (Theorem 5). We then
+use the local unitary result to establish a bounded
+but non-linear optimization program that deter-
+mines the optimal achievable fidelity under conver-
+sion via local operations and shared randomness
+(LOSR), which does not require shared randomness
+(Theorem 6). We end the section by discussing the
+relationship between the LU and LOSR strategies
+and introducing an SDP relaxation for efficiently es-
+tablishing upper bounds on the achievable fidelity
+of pure state conversions under LOSR.
+A.
+Correspondence Under Local Unitaries between
+Schmidt Coefficients and the Probability Simplex
+In this subsection we establish the bijection be-
+tween Schmidt coefficients, which define the equiv-
+alence classes of bipartite pure states under local
+unitaries, and the probability simplex. One reason
+for this is because the rest of the results of this work
+might be best seen as verifying that in the zero com-
+munication setting this correspondence is all that
+matters. Indeed, we will see this in the subsequent
+subsections which show that the minimal fidelity
+error of pure state transformations under zero com-
+munication will always be functions of only the
+Schmidt coefficients.
+Proposition 4. Up to local unitaries, any pure quan-
+tum state is of the form
+|ψ⟩AB = ∑
+i∈[k]
+�
+p↓(i) |i⟩A ⊗ |i⟩B ,
+where p↓(i) ≥ p↓(i + 1) for all i ∈ [k − 1], k =
+max{d, d′}, p↓ ∈ P↓(k), and {|i⟩} is the computa-
+tional basis in both cases. In other words, there exist
+both equivalence classes on pure states under local
+unitary operations in terms of Schmidt coefficients
+and ordered Schmidt coefficients.
+Proof. Consider |ψ⟩AB = ∑j∈[k]
+�
+p′(j)
+��uj
+� ⊗
+��vj
+�
+as
+decomposed in (4). Now fix the permutation π on
+[k] such that p′(π−1(i)) ≥ p′(π−1(i + 1)) for all
+i ∈ [k − 1], i.e. π re-labels p′ so that it is decreasing.
+Define the unitaries UA = ∑j∈[k] |π(j)⟩
+�
+uj
+��, WB =
+∑j∈[k] |π(j)⟩
+�
+vj
+��, which may be verified to be uni-
+taries by direct calculation. Then (UA ⊗ WB) |ψ⟩AB
+
+7
+will be of the form given in the proposition state-
+ment. Finally, we could make this argument for any
+pure state without ordering the Schmidt coefficients
+to get one set of equivalence classes. As such, under
+local unitaries, we can define equivalence classes
+of pure states in terms of ordered or non-ordered
+Schmidt coefficients. This completes the proof.
+Definition 1. The space of (representatives of the
+equivalence class of) ordered Schmidt coefficient
+pure states with Schmidt rank bounded by d is
+given by SR↓(d).
+That is, if |ψ⟩ ∈ SR↓(d), then
+|ψ⟩ = ∑i∈[d]
+�
+p↓(i) |ui⟩ |i⟩ |i⟩ where p↓ ∈ P↓(d).
+We can use the previous proposition to relate the
+(ordered) probability simplex over d elements to
+to the equivalence classes of (ordered) Schmidt de-
+compositions with Schmidt rank bounded by d.
+Proposition 5. Consider the functions vec(√·) :
+L(Cd) → Cd ⊗ Cd and vec−1(·⊙2) : Cd ⊗ Cd →
+L(Cd) where ·⊙2 is the entry-wise square of a vec-
+tor. These functions define a bijection between P(d)
+(resp. P↓(d)) and the space of equivalence classes
+of Schmidt decompositions under local unitaries
+with Schmidt rank bounded by d (resp. the space
+SR↓(d).)
+Proof. We prove it via direct calculation for P(d)
+and the space of Schmidt decompositions.
+The
+proof in the other case works the same. Let C ∼= Cd.
+First, consider p ∈ P(d) which we write in its den-
+sity matrix form, e.g. P = ∑i∈[d] p(i) |i⟩⟨i|. Then
+vec(
+√
+P) = vec
+�
+� ∑
+i∈[d]
+�
+p(i) |i⟩⟨i|
+�
+�
+= ∑
+i∈[d]
+�
+p(i) |i⟩C ⊗ |i⟩C′ ,
+which is in the specified equivalence class by ap-
+plying an isometries that take the computational
+bases from C, C′ to A, B. In the other direction, take
+the Schmidt decomposition in the purified basis,
+|ψ⟩AB = ∑i∈[d]
+�
+q(i) |i⟩A ⊗ |i⟩B. We can convert
+the A space to C via the channel
+FA→C(·) := V† · V + (1 − V†V) · (1 − V†V) ,
+where V = ∑i∈[d] |i⟩A ⟨i|C is the isometry that takes
+the C space to the A space as |A| ≥ |C| by assump-
+tion. The same type of conversion holds for the B
+and C′. Therefore, we have (up to equivalences)
+|ψ⟩AB = ∑i∈[d]
+�
+q(i) |i⟩C |i⟩C′. Then,
+vec−1(|ψ⟩·2) = vec−1( ∑
+i∈[d]
+q(i) |i⟩C |i⟩C′)
+= ∑
+i∈[d]
+q(i) |i⟩⟨i|C ,
+where in the last line we used that C′ ∼= C so that
+L(C, C′) ∼= L(C). This completes the proof.
+The reason this is useful is it draws equivalence
+between the equivalence classes of entangled states
+in terms of Schmidt coefficients and probability dis-
+tributions under fidelity.
+Proposition
+6.
+Consider
+|φ⟩
+=
+∑i∈[d]
+�
+p(i) |i⟩A |i⟩B, |ψ⟩ = ∑i∈[d]
+�
+q(i) |i⟩A |i⟩B.
+Then F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2.
+Proof. First note V : |i⟩A → |i⟩A |i⟩B is an isom-
+etry.
+Thus by isometric equivalence of fidelity
+(Item 2 of Proposition 1), we have F(|φ⟩ , |ψ⟩) =
+F(V |φ′⟩ V†, V |ψ′⟩ V†) where the primed versions
+just remove the B register. Then using Item 6 of
+Proposition 1 completes the proof.
+Randomness Embezzling
+Before moving forward,
+we note that independent of the focus of this work,
+this equivalence between Schmidt coefficients and
+the probability simplex means that the proof of
+quantum embezzlement also proves the existence
+of a classical version.
+Specifically, if one looked
+at the proof of quantum embezzlement [12], one
+would only need to note the starting and ending
+state they bound the fidelity between are in the
+computational basis locally and use
+F(|ψ⟩ , |φ⟩) = |⟨ψ, φ⟩|2 =
+���⟨
+√
+P,
+�
+Q⟩
+���
+2
+=
+�
+∑
+i
+�
+p(i)q(i)
+�2
+=BC(p, q)2 ,
+which follows the same argument as the previous
+few propositions, to ultimately conclude the same
+proof bounds a classical equivalent (Theorem 2). As
+we did not present the proof for embezzlement of
+quantum states, we present the proof of embezzle-
+ment of probability distributions in full for clarity
+in Appendix A.
+Theorem 2. For any ε > 0 and target probabil-
+ity distribution P ∈ P(m), the catalyst distribution
+Rn :=
+1
+Hn ∑n
+j=1
+1
+j |j⟩⟨j| is such that for n > m1/ε there
+exists a unitary representation of a basis relabeling
+Uf of the joint distribution such that
+F(Uf (Rn ⊗ |0⟩⟨0|)U†
+f , Rn ⊗ P) ≥ 1 − ε .
+
+8
+(a)
+(b)
+FIG. 3: Comparison between embezzlement of classical
+distributions and quantum states. (a) The embezzlement
+of classical distributions happens within one lab and a
+local permutation of the joint computational basis. (b)
+The embezzling of quantum states happens across two
+labs where each party applies the permutation of the
+joint computational basis on their local halves.
+We note the major difference between random-
+ness and quantum embezzlement is the role of lo-
+cality. In the classical case there is a single party
+and the distribution is not bipartite, both of which
+remove the notion of locality. These differences are
+non-trivial: one cannot construct a non-local classi-
+cal equivalent of embezzling that at the same time
+demands that the catalyst remains decoupled as
+in Proposition 3, and one cannot find a quantum
+equivalent of the non-local classical variation that
+one can implement as follows from Proposition 2.
+As it is not central to the rest of this work, we pro-
+vide an extended discussion of this nuance for the
+interested reader in Appendix A after the proof of
+Theorem 2.
+B.
+Pure State Conversion under Local Unitaries
+Having established the relationship between the
+equivalence classes of pure states in terms of
+Schmidt coefficients and the probability simplex,
+we now show the optimal strategy for converting
+one pure state to another under local unitaries is
+simply re-labeling the Schmidt basis so the order-
+ing of the Schmidt coeffficients is the same. This is
+not necessarily surprising. It is not clear what more
+one could do, and indeed this is the strategy that
+is used to implement quantum embezzlement [12].
+For intuition, we quickly show the equivalent result
+in the classical setting first.
+Proposition 7. Let p↓, q↓ ∈ P↓(d) Then for any i ∈
+[d] and d − k ≥ k ≥ 1,
+1 ≥
+�
+p↓(i)
+�
+q↓(i) ≥
+�
+p↓(i)
+�
+q↓(i + k) .
+Proof. This just follows from the fact if 1 ≥ p↓(i) ≥
+p↓(i + 1) and the same for q↓.
+Corollary 1. Given p, q ∈ P(d),
+max
+π∈Sd
+BC(p, πq) = BC(p↓, q↓) ,
+where Sd is the set of permutations on d elements.
+Proof. All we are looking for is the permutation of
+the elements of q such that BC(p, πq) is maximized.
+We can apply the permutation σ such that σPσ† =
+P↓, the matrix representation of p↓, to both sides. By
+the isometric equivalence of fidelity and that per-
+mutations are group, the problem is the same. That
+is, we can consider
+max
+π∈Sd
+BC(p↓, πq) = max
+π∈Sd ∑
+i∈[d]
+�
+p↓(i)
+�
+q(π(i)) .
+It immediately follows from Proposition 7 that the
+optimal π is the one that takes Q to Q↓. This com-
+pletes the proof.
+The idea is then to lift this result to quantum
+states optimized over unitaries and then use this
+with Uhlmann’s theorem to lift to the bipartite set-
+ting with local unitaries.
+The main challenge is
+minimizing over unitaries in the lift of the pre-
+vious result as now we have to deal with non-
+commutivity. This is done by reducing optimizing
+over unitaries to optimizing over permutations us-
+ing the Birkhoff-von Neumann Theorem.
+Lemma 3 (Birkhoff-von Neumann Theorem). Let
+d ∈ N. Given a linear operator X ∈ L(Rd), X is bis-
+tochastic (non-negative entries such that each col-
+umn and each row sums to one) if and only if there
+exists a probability distribution p ∈ P(|Sd|) such
+that
+X = ∑
+π∈Sd
+p(π)Vπ ,
+where Vπ(i, j) := δi,π(j) are permutation matrices
+and δi,j is the Kronecker delta. That is, a linear oper-
+ator is bistochastic if and only if it is a convex com-
+bination of permutation matrices.
+Lemma 4. Let ρ, σ ∈ D(Cd). Then
+max
+U
+F(ρ, UσU†) = F(P↓, Q↓) ,
+where P↓ = ∑i νi(ρ) |i⟩⟨i| and likewise for Q↓ but
+with respect to σ. In other words, the fidelity be-
+tween ρ and σ maximized over unitaries is equal to
+the fidelity of their ordered eigenvalues.
+Proof. First, by the isometric invariance of fidelity
+(Item 2 of Proposition 1), F(ρ, σ) = F(P↓, VσV†)
+
+P
+10)(0|
+Ur
+RnA
+A
+UAA'
+A'
+A'
+B'
+B'
+M
+BB'
+B
+B
+89
+where V is the unitary such that VρV† = P↓ =
+∑i νi(ρ) |i⟩⟨i|. As unitaries are closed under multi-
+plication and conjugate transpose,
+max
+U
+F(ρ, UσU†) = max
+U′ F(P↓, U′VσV†U′†)
+as the optimal U′ = U⋆V† where U⋆ is the opti-
+mizer for the L.H.S. of the equality. Therefore we
+just define Q ≡ VσV† and focus on solving the
+R.H.S. for clarity. Therefore, we are interested in
+maxU F(P↓, UQU†).
+Denote the spectral decomposition of Q
+=
+∑j q(j)
+��φj
+��
+φj
+��.
+Note that without loss of gener-
+ality, we may write U = ∑j
+��ψj
+� �
+φj
+�� for some
+orthonormal basis {
+��ψj
+�}.
+Therefore, UQU† =
+∑j q(j)
+��ψj
+��
+ψj
+��.
+Furthermore, define P(X)
+:=
+∑i |i⟩⟨i| X |i⟩⟨i|, which is the pinching, or dephasing,
+channel onto the computational basis. Then
+P(UQU†) =∑
+i,j
+|i⟩⟨i| q(j)
+��ψj
+��
+ψj
+�� |i⟩⟨i|
+=∑
+i,j
+q(j)|
+�
+i
+��ψj
+� |2 |i⟩⟨i| .
+Note that in contrast, P↓ is invariant under this
+pinching. Combining these points,
+max
+U
+F(P↓, UQU†)
+≤ max
+U
+F(P↓, P(UQU†))
+= max
+U
+Tr
+��√
+P↓P(UQU†)
+√
+P↓
+�1/2�2
+= max
+{|ψj⟩}
+Tr
+��
+∑
+j,i,i′,�i
+q(j)
+�
+p↓(i′)p↓(�i)
+���
+�
+i
+��ψj
+� ���
+2
+·
+��i′� �
+i′��i
+� �
+i
+����i
+� �
+�i
+���
+�1/2
+�2
+= max
+{|ψj⟩}
+Tr
+�
+�
+�
+∑
+j,i
+q(j)p↓(i)|
+�
+i
+��ψj
+� |2 |i⟩⟨i|
+�1/2�
+�
+2
+,
+where the inequality is the data-processing inequal-
+ity (Item 3 of Proposition 1) with the pinching chan-
+nel along with the invariance of P↓ under this chan-
+nel, the first equality is using the definition of fi-
+delity (2), the second is just expanding everything,
+and the third is collapsing the implicit Kronecker
+deltas.
+Now note the following trick.
+We can define
+the square matrix A via its elements: A(j, i) :=
+|
+�
+i
+��ψj
+� |2.
+We know 0
+≤
+|
+�
+i
+��ψj
+� |2
+≤
+1,
+∑i |
+�
+i
+��ψj
+� |2 = 1, and ∑j | ⟨i|j⟩ |2 = 1 as {|i⟩} and
+{
+��ψj
+�} are orthonormal bases. It follows that A is
+a bistochastic matrix by definition. Therefore, by
+the Birkhoff-von Neumann Theorem (Lemma 3),
+A = ∑π∈Sd r(π)Wπ where Wπ is the permutation
+matrix for π and r is a probability distribution over
+the permutations. Thus, plugging this back in to
+what we started with,
+max
+{|ψj⟩}
+Tr
+�
+�
+�
+∑
+j,i
+q(j)p↓(i)|
+�
+i
+��ψj
+� |2 |i⟩⟨i|
+�1/2�
+�
+2
+= max
+r
+Tr
+�
+�
+�
+∑
+j,i,π
+q(j)p↓(i)r(π)Wπ(j, i) |i⟩⟨i|
+�1/2�
+�
+2
+.
+Now note every permutation matrix is the iden-
+tity matrix with columns permuted, i.e.
+π
+=
+�
+eT
+π(0) eT
+π(1) . . . eT
+π(d−1)
+�T
+, where ei := |i⟩. It fol-
+lows that
+∑
+π∈Sd
+r(π)Wπ(j, i)
+=∑
+π
+r(π)1{Wπ(j, i) = 1}
+=∑
+π
+r(π)1{π(j) = i} =: Pr
+r [π(j) = i′] ,
+where 1{A} is the indicator function for an event
+and the second equality is because W(j, i) = 1 if
+and only if π(j) = i. We stress the final definition
+is a function of the choice of r and j, i and form a
+joint probability over (j, i) as ∑j Prr[π(j) = i] = 1 =
+∑i Prr[π(j) = i] and every element is non-negative.
+This simplifies the problem to
+max
+r
+Tr
+�
+�
+�
+∑
+j,i,π
+q(j)p↓(i)r(π)Wπ(j, i) |i⟩⟨i|
+�1/2�
+�
+2
+= max
+r
+�
+∑
+i
+�
+p↓(i)
+�
+���
+j
+�
+q(j) Pr
+r [Π(j) = i]
+��2
+,
+where we have just grouped terms and used that
+the operator is diagonal, so we can apply the square
+root entry-wise and take the sum to compute the
+trace.
+So we want to determine the maximal distri-
+bution r, but we can show this is achieved by
+element-wise optimizing the sum.
+Note
+�
+p↓(1)
+is the largest element and bounded above by 1,
+so we want to multiply it by the largest value
+∑j
+�
+q(j) Prr[π(j) = i] can take.
+�
+q(j) ≤ 1 for
+all j and ∑j Prr[π(j) = i] = 1 so the largest value
+
+10
+this sum can take is maxj
+�
+q(j). Note if we pick
+a different distribution each term will be smaller
+than it could be by Proposition 7. This means we
+choose r such that all non-zero probability permu-
+tations map argmaxj q(j) to 1. We then have the
+same problem as initially but with
+�
+p↓(2) serving
+the largest element and q not containing its largest
+element. Doing the argument recursively, we con-
+clude the optimal distribution r has unit probability
+on permutation σ such that ∑i q(σ(i)) = q↓. Thus,
+max
+r
+�
+∑
+i
+�
+p↓(i)
+�
+∑
+j
+�
+q(j) Pr
+r [Π(j) = i]
+��2
+=
+�
+∑
+i
+�
+p↓(i)
+�
+q↓(i)
+�2
+=BC(p↓, q↓)2
+=F(P↓, Q↓),
+where the first equality is by the preceding explana-
+tion and the last two are using Item 5 of Proposition
+1. Note this means we have established an upper
+bound as we used the data processing inequality at
+the beginning. However, this is clearly achievable
+by picking by the permutation unitary that maps σ
+to Q↓. Thus this completes the proof.
+We now can use the above lemma to establish the
+pure state property we are actually interested in.
+For notational simplicity, we define the following
+notation:
+FLU(ρ, σ) := max
+U,V F(ρ, (U ⊗ V)(σ)) ,
+which is without loss of generality unitaries as we
+can just trivially embed the smaller dimensional
+state.
+Theorem 5.
+FLU(|ψ⟩ , |φ⟩) = F(P↓, Q↓) ,
+where P↓ is the distribution defined by the decreas-
+ing Schmidt coefficients of |ψ⟩ and likewise for Q↓
+and |φ⟩. In other words, the optimal fidelity of con-
+verting |φ⟩ to |ψ⟩ via local unitaries is given by the
+fidelity of their ordered Schmidt coefficients.
+Proof. Up to local unitaries, |ψ⟩ = ∑i
+�
+p↓ |i⟩ |i⟩.
+Therefore without loss of generality, that can be
+taken as our target state by allowing free local uni-
+taries on the seed state. We can take the seed state
+to be of the form |φ⟩ = ∑i
+�
+q(i) |i⟩ |i⟩ by the same
+argument. Then by assumption, we are interested
+in maxU,V F(|ψ⟩ , (U ⊗ V) |φ⟩) with the specified
+forms. Note
+TrB((U ⊗ V) |φ⟩⟨φ| (U ⊗ V)†)
+=∑
+i,i′
+�
+q(i)q(i′)U |i⟩
+�
+i′�� U† Tr
+�
+V |i⟩
+�
+i′�� V†�
+=∑
+i
+q(i)U |i⟩⟨i| U† =: UQU†.
+Now for any unitary U we define the following pu-
+rification
+���w|U�
+:= vec(
+�
+UQU†)
+=(U ⊗ U) vec(
+�
+Q) = (U ⊗ U) |φ⟩ ,
+where we have used
+�
+UQU† = U√QU† and the
+vec map identity (1). Now we have
+F(P↓, UQU†) = max
+|w′⟩ F(|ψ⟩ ,
+��w′�)
+= max
+V
+F
+�
+|ψ⟩ , (1 ⊗ V)
+���w|U��
+= max
+V
+F(ψ, (U ⊗ VU) |φ⟩) ,
+(6)
+where the first equality is by Uhlmann’s theorem
+(Lemma 1), the second is because all purifications
+of a given operator are unitarily equivalent on the
+purifying space, so there exists a V such that (1 ⊗
+V)
+���w|U�
+= |w′⟩. The final line is just expanding
+the definition of
+���w|U�
+.
+It follows,
+max
+W,V F(|ψ⟩ , (W ⊗ V) |φ⟩)
+= max
+U,V′ F(|ψ⟩ , (U ⊗ V′U) |φ⟩)
+= max
+U,V′ F(|ψ⟩ , (1 ⊗ V)
+���w|U�
+)
+= max
+U
+F(P↓, UQU†)
+=F(P↓, Q↓) ,
+where the first equality is because unitaries are
+closed under multiplication and the optimizations
+are independent, the second and third are both
+by (6) for clarity, the third is because unitaries are
+closed under conjugation and then the final equal-
+ity is by applying Lemma 4.
+This completes the
+proof.
+This means under local unitaries, it is efficient to
+compute the optimal fidelity and that in fact the op-
+timal strategy is simply Alice and Bob re-ordering
+the basis so that the Schmidt coefficients are in the
+
+11
+same relative ordering. It also follows from Item
+1 of Proposition 1 that unless all the Schmidt coef-
+ficients are equal, the fidelity cannot be one under
+local unitary strategies.
+C.
+Pure State Conversions under Local Operations
+and Shared Randomness
+While the previous section is nice in that it finds
+an efficient way of calculating the optimal conver-
+sion strategy under local unitaries, it would be nat-
+ural to ask if local operations can do better than lo-
+cal unitaries as it is a much more general class of
+operations. In fact, we can see that it must do bet-
+ter in some cases in a trivial manner. Consider the
+target state |ψ⟩ and the seed state |φ⟩ = |ψ⟩ ⊗ |ζ⟩
+where |ζ⟩ is not product. Under local unitaries this
+transformation isn’t possible to arbitrary precision
+because of |ζ⟩, but of course in reality the parties
+could trace out whichever portion(s) of |ζ⟩ they
+hold. Thus, we need a theory of transformations
+under local operations.
+Note that this trivial example we have given
+would not be resolved by local mixed unitary
+strategies. Indeed, we begin by noting that local
+mixed unitary strategies cannot ever outperform lo-
+cal unitary strategies.
+Corollary 2. Let |ψ⟩ be the target state and |φ⟩ be
+the seed state and only optimize over Alice and Bob
+using mixed unitary channels. Then the optimal is
+the same as in Theorem 5.
+Proof. Letting EU, FW be local mixed unitary maps,
+max
+EU,FW
+F(ψ, (EU ⊗ FW)(φ))
+= ⟨ψ| (EU ⊗ FW)(φ) |ψ⟩
+=
+�
+U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩ dU dW
+≤
+�
+U,W max
+U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩
+= max
+U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩
+=F(P↓, Q↓) ,
+where the first equality is by Item 4 of Proposition 1,
+the second is letting the mixed unitary map be for
+any probability measures dU,dW over the unitary
+group. The inequality is because the inner product
+is real and so it is lower bounded by the maximum.
+The second to last equality is by linearity, and the
+final equality is by Theorem 5. Noting that a specific
+choice of local unitaries is a special case of mixed
+unitary channels completes the proof.
+The above tells us that we must escape the use
+of unitaries to improve our bounds. Note however
+that in general the only maps that preserve pure
+states are isometries, and our results so far have
+been in terms of pure states, so we need to main-
+tain this structure to build on them. For this reason,
+the following proof will make use of the isometric
+representation of quantum channels.
+For notational simplicity, we define the optimal
+fidelity of conversion under local operations and
+shared randomness (LOSR) fidelity
+FLOSR(ρ, σ) := max
+µ,Eλ,Fλ
+F(ρ,
+�
+(Eλ ⊗ Fλ)(σ)dµ(λ)) ,
+where µ is a probability measure over an index set
+for sets of local channels {Eλ} and {Fλ}. Similarly,
+we can define optimal fidelity of conversion under
+local operations (LO) as
+FLO(ρ, σ) := max
+E,F F(ρ, E ⊗ F)(σ)) .
+With these defined, we prove the following.
+Theorem 6.
+FLOSR(|ψ⟩ , |φ⟩) = FLO(|ψ⟩ , |φ⟩)
+=
+max
+P′∈P(Σ) F((P ⊗ P′)↓, Q↓
+embed) ,
+where |Σ| ≤ SR(|φ⟩) · SR(|ψ⟩), P is the probability
+distribution defined by |ψ⟩’s Schmidt coefficients
+and likewise for Qembed with the Schmidt coeffi-
+cients of |φ⟩ except the distribution is embedded
+into the joint space.
+Proof. The first equivalence follows similarly to the
+mixed unitary case.
+Clearly the class of LOSR
+strategies is more general than the class of LO
+strategies, so we just need to show LOSR is only
+as strong as LO here.
+FLOSR(φ, ψ) =F
+�
+ψ,
+�
+(Eλ ⊗ Fλ)(φ)dµ(λ)
+�
+=
+�
+⟨ψ| (Eλ ⊗ Fλ)(φ) |ψ⟩ dµ(λ)
+≤
+�
+max
+E,F [⟨ψ| (E ⊗ F)(φ) |ψ⟩] dµ(λ)
+= max
+E,F ⟨ψ| E ⊗ F)(φ) |ψ⟩
+=FLO(φ, ψ) ,
+where the first equality is by definition and denot-
+ing the optimizers by µ, {Eλ}, {Fλ}, the second is
+by linearity of the Lebesgue integral, the inequal-
+ity is because ⟨ψ| (E ⊗ F)(φ) |ψ⟩ is a real number
+for any choice of local channels, the third equality
+
+12
+is because µ is a probability measure that is now
+independent of the argumenbt of the integral, and
+the final equality is by definition. This proves the
+reduction of LOSR to LO if the target state is pure.
+Next, we bound the dimension of Σ. We want
+to consider maxE,F F(ψ, (E ⊗ F)(φ)). Without loss
+of generality, we assume the local spaces are ‘com-
+pressed’ such that din := SR(|φ⟩) so that E, F both
+act on L(Cdin). We now show that without loss of
+generality we may restrict the output dimension of
+E, F to be dout := SR(|ψ⟩).
+This is just because
+we can project onto the support of the marginal
+of |ψ⟩ on both local spaces, so we can restrict the
+local maps to this space.
+Formally, this can be
+seen as follows.
+Consider arbitrary E, F and let
+|ψ⟩ = ∑i
+�
+p(i) |i⟩ |i⟩. Define ΠP := ∑i:p(i)>0, i.e.
+the projector onto the support of TrB(ψ) = TrA(ψ),
+where the equality is up to the change in space.
+Note rank(ΠP) = Schmidt(ψ).
+By construction,
+(ΠP ⊗ ΠP) |ψ⟩ = |ψ⟩. Therefore,
+F(ψ, (E ⊗ F)(φ))
+= ⟨ψ| (E ⊗ F)(φ) |ψ⟩
+= Tr[|ψ⟩⟨ψ| (E ⊗ F)(φ)]
+= Tr
+�
+ψΠ⊗2
+P (E ⊗ F)(φ)Π⊗2
+P
+�
+,
+where in the first equality we have used Item 4
+of Proposition 1 and the other two use cyclicity of
+trace along with invariance of ψ under the projec-
+tor. Now we can expand,
+Π⊗2
+P (E ⊗ F)(φ)Π⊗2
+P
+=∑
+k,l
+ΠPAk ⊗ ΠPBkφA†
+kΠP ⊗ B†
+l ΠP
+≡(EΠ ⊗ FΠ)(ψ) ,
+where {Ak}, {Bl} are the Kraus operators of E, F
+respectively and EΠ, FΠ are CPTNI maps defined
+by {ΠPAk}, {ΠPBl} respectively. Note this equiva-
+lence holds as (ΠAk)† = A†
+kΠP since Π†
+P = ΠP so
+it is CP and it is TNI because
+∑
+k
+(ΠPAk)†(ΠPAk) =∑
+k
+A†
+kΠPAk
+≤∑
+k
+A†
+k1Ak = 1 ,
+where we used Π2
+P = ΠP in the first equality,
+ΠP ≤ 1 and that E is CP in the inequality, and
+that E is TP in the last inequality. An identical ar-
+gument holds for FP. This proves the optimizer
+is achieved with CPTNI maps T(L(Cdin), L(Cdout)).
+Finally, we can lift EP, FP to being CPTP, denoted
+�E, �F ∈ T(L(Cdin), L(Cdout)) by adding one Kraus
+operator, e.g. for EP add the Kraus operator Z ∈
+L(Cdin, Cdout) where Z†Z = (1 − ∑k A†
+kΠAk) ≥ 0
+which always exists by definition of the space of
+positive semidefinite operators. By linearity,
+F(ψ, (E ⊗ F)φ) = Tr[ψ(EΠ ⊗ FΠ)(φ)]
+≤ Tr
+�
+ψ( �E ⊗ �F)(φ)
+�
+.
+Therefore without loss of generality the optimal
+channels are E, F ∈ C(Cdin, Cdout). Note this means
+that Rank(JE) ≤ dindout and likewise for JF.
+We now derive the equation using the isometric
+representation of the channel.
+max
+E,F F(ψ, (E ⊗ F)(φ))
+=⟨ψ, (E ⊗ F)(φ)⟩
+= max
+V1,V2,|ζ⟩ |⟨ψ| ⟨ζ| (V1 ⊗ V2) |φ⟩|2
+=
+max
+U1,U2,|ζ⟩
+���⟨ψ| ⟨ζ| (U1 ⊗ U2) |φ⟩ |0⟩E1 |0⟩E2
+���
+2
+=
+max
+U′
+1,U′
+2,
+���ζp′
+�
+���⟨ψ|
+�
+ζp′
+��� (U′
+1 ⊗ U′
+2) |φ⟩ |0⟩E1 |0⟩E2
+���
+= max
+P′
+F((P ⊗ P′)↓, Q↓
+embed) ,
+where the second line is because there exists an iso-
+metric representation of each channel which means
+(V1 ⊗ V2)(φ) is a pure state, so we can apply Uhlm-
+man’s theorem to find a purification of |ψ⟩ that sat-
+urates the bound, but as |ψ⟩ is already pure, any
+purification will be a product state. The third line
+is because we can always convert an isometry into
+a unitary on the appropriately large space.
+The
+fourth line means that ζp′ = ∑i′
+�
+p′(i) |i⟩ |i⟩, which
+can always be achieved by local unitaries on the
+E1 and E2 spaces, which result on new unitaries
+on the other side but the same maximum. The fi-
+nal equality is just using Theorem 5 and we write
+Qembed to stress it is defined over the whole alpha-
+bet. Lastly, as we established bounds on the ranks
+of the local maps Choi matrices, we have bounds
+E1, E2 ≤ dindout, which justifies the maximum and
+tells us how large of a system we have to consider
+in the statement of the theorem.
+It is useful to see how this result works. It in ef-
+fect shows the following equivalence of conversion
+when measured under fidelity
+|φ⟩ −→
+LO |ψ⟩ = max
+|ζ⟩
+�
+|φ⟩ −→
+LU |ψ⟩ ⊗ |ζ⟩
+�
+,
+which can be viewed both by proof and via intu-
+ition as a special case of the isometric representa-
+tion of a channel. Moreover, it is easy to see in this
+
+13
+form how it handles our motivating example. In-
+deed, if the target state is |ψ⟩ and the seed state is
+|ψ⟩ ⊗ |ζ⟩, then clearly the maximizer is chosen by
+the ancillary state being |ζ⟩ and the local unitaries
+being trivial.
+D.
+Relation between LO and LU Strategies
+The natural question given the previous theo-
+rems is if we can better understand the relationship
+between LO and LU strategies. We first show that
+LU and LO strategies are equivalent when either
+the target or the seed state is a two qubit state.
+1.
+LU and LO Equivalence for Two-Qubit Seed or Target
+State
+Proposition 8. Consider entangled two qubit seed
+state |φ⟩ ∈ C2 ⊗ C2.
+Let the target entangled
+state be |ψ⟩ ∈ Cd ⊗ Cd′.
+Then the optimal non-
+communicative strategy is the local unitary strat-
+egy.
+Proof. Without loss of generality, q↓ = (q, 1 − q)
+where q ≥ 1/2 and p↓ = (p(1), p(2), ...).
+Then
+the optimal local unitary strategy is
+�
+qp(1) +
+�
+(1 − q)p(2).
+For any P′ we can write (p′)↓ =
+(p′(1), p′(2), ...). The optimal CPTP strategy (up to
+a square) is of the form
+�
+qp(1)p′(1) +
+�
+(1 − q) max{p(1)p′(2), p(2)p′(1)} .
+These values can only increase by assuming p′ has
+two outcomes, so let us assume so without loss
+of generality and parameterize the distribution by
+p′ ∈ [1/2, 1] to obtain
+�
+qp(1)p′ +
+�
+(1 − q) max{p(1)(1 − p′), p(2)p′} .
+Moreover note p(2)p′ < p(2) unless p′ = 1, which
+is equivalent to the LU strategy, so the second entry
+in the maximization would be lower than the LU
+setting. Therefore, we focus on the remaining case.
+We are specifically interested in when the following
+strict inequality holds:
+�
+qp(1)p′ +
+�
+(1 − q)p(1)(1 − p′)
+>
+�
+qp(1) +
+�
+(1 − q)p(2)
+⇔ g(p′) :=
+�
+qp(1)(
+�
+p′ − 1)
++
+�
+1 − q(
+�
+p(1)(1 − p′)
+−
+�
+p(2)) > 0 .
+Then
+d
+dp′ g(p′) =
+√
+qp(1)
+2√
+p′ +
+√
+p(1)(1−q)
+2√
+1−p′
+. It follows,
+�
+qp(1)
+�
+1 − p′
+2
+�
+p′�
+1 − p′
++
+�
+p′�
+p(1)(1 − q)
+2
+�
+1 − p′�
+p′
+≥ 0
+⇔
+�
+qp(1)
+�
+1 − p′ +
+�
+p′
+�
+p(1)(1 − q) ≥ 0
+⇔√q
+�
+1 − p′ +
+�
+p′
+�
+(1 − q) ≥ 0
+⇔
+√
+F(Q↓, P′↓) ≥ 0 ,
+where the first line is multiplying to get identical
+denominators, the second line is multiplying by the
+denominator, the third is dividing out p(1), and
+the final is by the definition of square root fidelity.
+Note the final inequality will always hold strictly
+unless q ∈ {0, 1}, i.e. the state is a product state,
+by Item 1 of Proposition 1.
+If q ∈ {0, 1}, then
+the state is a product state which would contradict
+that we assume the state is entangled. Therefore,
+in our setting, g(p′) only increases over its inter-
+val, p′ ∈ [0, 1]. Thus, the optimal choice of p′ is
+p′ = 1, but in this case the value is
+�
+qp(1) ≤
+�
+qp(1) +
+�
+(1 − q)p(2), i.e. the optimal choice is
+lower bounding the optimal local unitary strategy.
+It follows this is never optimal. This completes the
+proof.
+Proposition 9. Consider entangled two qubit target
+state |ψ⟩ ∈ C2 ⊗ C2 and any seed state |φ⟩ ∈ Cd ⊗
+Cd′. The optimal non-communicative strategy is the
+local unitary strategy.
+Proof. The proof is basically the same as for the two
+qubit seed case. Without loss of generality, p↓ =
+(p, 1 − p). We can re-order |φ⟩ such that it is q↓ =
+(q(1), q(2), ...). The optimal CPTP strategy (up to a
+square) is of the form
+�
+q(1)p′(1)p +
+�
+q(2) max{p′(1)(1 − p), p′(2)p} .
+This sum can only increase if p′(1) + p′(2) = 1,
+so we can parameterize the distribution by p′ ∈
+[1/2, 1] to obtain
+�
+q(1)p′p +
+�
+q(2) max{p′(1 − p), (1 − p′)p} .
+Note that p′(1 − p) < (1 − p) unless p′ = 1. If
+p′ = 1, this is the LU strategy, if p′ < 1, then this is
+worse than an LU strategy. Therefore, we only care
+about the other maximization case. That is, we are
+
+14
+interested in when p ∈ [1/2, 1) and the following
+strict inequality holds:
+�
+q(1)p′p +
+�
+q(2)p(1 − p′)
+>
+�
+q(1)p +
+�
+q(2)(1 − p) .
+However,
+�
+q(1)p′p
+<
+�
+q(1)p
+and
+�
+q(2)p(1 − p′) <
+�
+q(2)(1 − p) as p′ ∈ [1/2, 1).
+Therefore this strict inequality can never hold.
+Therefore the optimal strategy is always the LU
+strategy. This completes the proof.
+2.
+LU and LO Inequivalence for States with Schmidt Rank
+Greater than Two
+If there is equivalence for two qubit seed or tar-
+get states, it is natural to ask if this property per-
+sists. One might expect that this is a special prop-
+erty of qubit systems as are found throughout quan-
+tum information science results. Indeed, generally
+this property does not hold, which we will prove
+via example.
+Theorem 7. For seed and target state with Schmidt
+rank ≥ 3, the optimal LO strategy may be better
+than the optimal LU strategy.
+Proof. We construct an example for Schmidt rank 3.
+By continuity of the fidelity, one can embed the tar-
+get and seed in bigger spaces with arbitrarily small
+perturbations for it to hold in higher dimensions,
+which is why this is sufficient. Consider target state
+|ψ⟩ = 0.85 |00⟩ + 0.08 |11⟩ + 0.07 |22⟩ and seed state
+|φ⟩ = 0.45(|00⟩ + |11⟩) + 0.1 |22⟩. Then, the optimal
+LU strategy fidelity is
+F(P↓, Q↓)
+=
+�√
+0.45(
+√
+0.85 +
+√
+0.08) +
+�
+0.1(0.07)
+�2
+<0.796 .
+In contrast, if we consider P′ = [0.55, 0.28, 0.17],
+then
+F((P ⊗ P′)↓, Q↓)
+=
+�√
+0.45
+√
+0.4675 +
+√
+0.45
+√
+0.238 +
+√
+0.1
+√
+0.1445
+�2
+>0.82 .
+As we maximize over P′, the optimal LO strategy
+achieves a value that is strictly above the LU strat-
+egy. This completes the proof.
+E.
+Inefficiency of Optimal LOSR Fidelity and
+Computable Upper Bounds
+In the above we have constructed an example
+where the local operations strategy outperforms the
+local unitary strategy (though we have not shown
+what the strategy itself is).
+A natural question
+would then be how easy it is to solve for the op-
+timal fidelity value or even a bound. By Theorem
+5, we can conclude the optimal local unitary strat-
+egy is polynomial time to solve as all one needs to
+do is sort the Schmidt coefficients and calculate the
+fidelity. Indeed, one could solve for the ordering of
+the Schmidt coefficients using the linear program
+for sorting a vector.
+In contrast, for optimizing LO strategies, we have
+no such luck. In effect this is because there are two
+things to optimize over at once. Indeed, recall
+FLO(|ψ⟩ , |φ⟩) =
+max
+P′∈P(Σ) F((P ⊗ P′)↓, Q↓) .
+Then the problem is that one must first tensor P
+onto variable P′ and then re-order the vector. One
+cannot even in general order an optimization vari-
+able, which we will refer to as ‘sorting,’ as sorting is
+in general non-convex. In sorting a vector using a
+linear program, one relaxes to bistochastic channels
+and considers a linear function so that the optimizer
+is an extreme point which by the Birkhoff von Neu-
+mann theorem is a specific permutation. However,
+we are many levels of involvement above that: we
+want the distribution P′ such that its product dis-
+tribution P ⊗ P′ when sorted optimizes the fidelity
+with Q↓. Therefore, we need to optimize over P′
+and the permutation at the same time. It’s not clear
+that we can actually relax to bistochastic strategies
+because of the joint concavity of fidelity. That is to
+say, for any bistochastic channel E,
+F(E(P ⊗ P′), Q↓) =F(∑
+π
+r(π)Vπ(P ⊗ P′), Q↓)
+≥∑
+π
+F(r(π)Vπ(P ⊗ P′), r(π)Q↓)
+=∑
+π
+r(π)F(Vπ(P ⊗ P′), Q↓) ,
+where the first line is Birkhoff-von Neumann the-
+orem, the second is joint concavity using Q↓ =
+∑π r(π)Q↓ as r is a probability distribution, and
+the last line is because F(λP, Q) = λF(P, Q) =
+F(P, λQ).
+Thus any bistochastic channel may
+strictly do better than the average of its extreme
+points. Moreover, even if we could optimize over
+bistochastic channels, we would have a non-convex
+objective function as the bistochastic channel, an
+optimization variable, would be applied to P ⊗ P′
+which is also partially an optimization variable.
+
+15
+Given the above, it seems likely the best option if
+one were to try and find a (near) optimum would be
+to use gradient descent from random initial P′, real-
+izing it will only work locally and will break down
+at ‘kinks’ where the ordering changes. Otherwise
+more sophisticated non-convex optimization tech-
+niques might be used.
+Computable Upper Bound Methods
+Perhaps even
+worse than our inability to calculate the exact fi-
+delity, is that it is not clear in general how to de-
+termine good bounds. Certainly we have the fol-
+lowing result.
+Theorem 8. Unless the target state is |ψ⟩ = |φ⟩ ⊗
+|ζ⟩ where |φ⟩ is the seed state, there exists ε > 0
+such that there does not exist local operations that
+will take |φ⟩ to |ψ⟩.
+Proof. This follows from Theorem 6 along with Item
+1 of Proposition 1.
+The above theorem, while derived from a very
+different strategy than Proposition 2, does not seem
+to give us much more information as to at what
+point communication is necessary. What we would
+want to efficiently improve this would be to estab-
+lish upper bounds on the equation given in Theo-
+rem 6 that have a closed form that does not depend
+on P′. One option is to use the data processing in-
+equality for fidelity. This can be seen in the follow-
+ing proposition.
+Proposition 10. Consider target state |ψ⟩ and seed
+state |φ⟩ with corresponding Schmidt distributions
+p, q respectively. If pmax ≤ qmax, then
+FLO(|ψ⟩ , |φ⟩) ≤ F(p, q) ,
+where p = pmax |0⟩⟨0| + (1 − pmax) |1⟩⟨1| and like-
+wise for q.
+Proof. Without loss of generality let d be the maxi-
+mum local dimension. Let E(·) = |0⟩⟨0| · |0⟩⟨0| +
+∑i∈{1,...,d−1} |1⟩ ⟨i| · |i⟩ ⟨1|. That is, E coarse-grains
+a probability distribution to the Bernoulli distribu-
+tion with its first element untouched and the sum
+of all the others as the other outcome. Then using
+data processing of fidelity (Item 3 of Proposition 1),
+max
+P′∈P(Σ) F((P ⊗ P′)↓, Q↓)
+≤ max
+P′∈P(Σ) F(E((P ⊗ P′)↓), E(Q↓))
+= max
+p′∈[0,1] F
+�
+�P(p′), E(Q↓)
+�
+,
+where �P(p′) := pmaxp′ |0⟩⟨0| + (1 − pmaxp′) |1⟩⟨1|
+and E(Q↓) = qmax |0⟩⟨0| − (1 − qmax) |1⟩⟨1|. Now
+note that by assumption pmax ≤ qmax. As the fi-
+delity will only decrease as pmaxp′ moves away
+from qmax, the optimal choice is p′ = 1. This com-
+pletes the proof.
+The problem with the above bound is that there
+will be cases where pmax > qmax.
+Why the in-
+equality in the other direction was required was to
+know for a fact what element of p was relevant,
+namely pmax and that any choice of p′ ̸= 1 would be
+sub-optimal. In general this strategy would require
+q↓(j) is sufficiently large relative to p↓(j). This can
+be determined in some cases. Here we provide a
+simple example.
+Example 3. Let
+p↓ = [3/4, 1/8, 1/8]T
+q↓ = [1/2, 1/2]T .
+Then (p ⊗ p′)↓[1 : 2] = p′(1)[3/4, 1/8]T, and so
+we can coarse-grain on the second element to ob-
+tain P(p′) = 1/8p′ |0⟩⟨0| + (1 − 1/8p′) |1⟩⟨1| and
+Q = Q↓. Then as 1/8p′ < 1/2, the upper bound
+is F( 1
+8 |0⟩⟨0| + 7
+8 |1⟩⟨1| , 1
+21) ≈ 0.83.
+The above shows that while data processing can
+be sufficient in certain cases, it does not provide
+an easy general method. Another common alter-
+native in quantum information theory is semidefi-
+nite relaxations of optimization problems because
+semidefinite programs are efficient to evaluate.
+In Appendix B, we establish the following upper
+bound and show it may be expressed as a semidef-
+inite program, which, as everything is in terms of
+probability distributions, is due to the non-linearity
+of fidelity and nothing particularly quantum.
+Theorem 9. Consider target state |ψ⟩ and seed state
+|φ⟩. Let SR(ψ) = d and SR(φ) = d′. Define A = Cd,
+B = Cd·d′. Then,
+FLOSR(|ψ⟩ , |φ⟩) ≤ max F(R, Q↓
+embed)
+s.t. TrB[R] = P↓
+R ∈ P↓(d2 · d′) ,
+(7)
+where P and Q are the distributions defined by |ψ⟩
+and |φ⟩’s Schmidt coefficients respectively. More-
+over, this admits the following simple semidefinite
+program over the reals:
+max ∑
+i∈[d2·d′]
+x(i)
+s.t.
+�diag(r)
+diag(x)
+diag(x) diag(q↓
+embed)
+�
+⪰ 0
+TrB[diag(r)] = P↓
+r ∈ P↓([d2 · d])
+x ∈ Rd2·d′ ,
+(8)
+
+16
+Physically, this relaxation may be seen as relaxing
+the isometric representation of the optimal LOSR
+strategy to one where one allows the ancillary en-
+vironment start off entangled with the local system.
+Mathematically, this is not too loose because we re-
+quire this entangled pure state has a notion of “lo-
+cal Schmidt coefficients” that pertain to the original
+target state, although this physically does not seem
+to have a clean interpretation. Nonetheless, we can
+see that (7) will not achieve unity unless there exists
+a joint distribution Q = R, which would require
+Q↓
+embed to have P↓ as it’s marginal, which seems
+highly restrictive. Therefore, (7) should provide an
+upper bound that is non-trivial.
+V.
+MANY COPY PURE STATE CONVERSION
+WITH ZERO COMMUNICATION
+Having established what happens for single
+copies,
+we consider many copies.
+We pro-
+vide two motivations for doing this.
+First, we
+note that it’s not clear what the limiting be-
+haviour will be even in the LU setting.
+A
+reader may recall from other works that the fi-
+delity is multiplicative so if F(P, Q) < 1, then
+limn→∞ F(P⊗n, Q⊗n)
+=
+limn→∞ F(P, Q)n
+→
+0.
+However, we lose the multiplicativity as we are
+considering limn→∞ F((P⊗n)↓, (Q⊗n)↓). This issue
+is further aggravated if we consider local opera-
+tions and the ancillary variable.
+The second motivation is that what was initially
+considered in the literature, albeit with LOCC [23],
+was the conversion of many copies of states. A par-
+ticular focus in the referenced work and subsequent
+ones is the case where either the target or seed state
+is the maximally entangled state, known as distil-
+lation and dilution respectively. With LOCC, we
+know there are ‘rates’ in the conversions. By [6]
+along with previous results in this work, we would
+not expect there to be non-negative rates without
+the communication assuming the error is required
+to be vanishing, i.e. ε → 0.
+In this section we establish convex optimiza-
+tion problems for dilution and distillation in the
+zero communication setting.
+These results are
+established in terms of the not-actually-a-norm
+∥ · ∥(k,1/2), which we remind the reader is the
+(k, p)−norms extended to p < 1 introduced in Sec-
+tion III with the choice of p = 1/2. We also look
+at the limiting behaviour as the number of copies
+grows. In particular, we find a closed form when
+trying to convert n−fold two qubit states to a dif-
+ferent n−fold two qubit state. Moreover, we prove
+the fidelity goes to zero in this case. We discuss
+the extension of this to entangled states with larger
+Schmidt rank.
+1.
+Dilution Under Local Operations
+We begin by determining the limits of dilution.
+For intuition, we begin with local unitaries where
+there is no optimization. Recall that the Schmidt
+coefficients of the maximally entangled state are all
+√
+d−1, so they correspond to the maximally mixed
+distribution under our bijection between Schmidt
+coefficients and probability distributions.
+Proposition 11. For local unitary strategies the op-
+timal dilution fidelity is given by
+FLU
+�
+|ψ⟩ ,
+��Φ+
+d
+�⊗n�
+= d−n ∥P∥(dn,1/2) .
+Proof. Generally, if |ψ⟩ ̸=
+��Φ+
+d
+�
+,
+1 >F(P↓, π⊗n
+d )↓)
+=F(P↓, π⊗n
+d )
+=
+�
+�d−n/2 ∑
+i∈[dn]
+�
+P↓(i)
+�
+�
+2
+=d−n
+�
+� ∑
+i∈[dn]
+�
+P↓(i)
+�
+�
+2
+=d−n∥P∥(dn,1/2) ,
+where the first equality is because π⊗n
+d
+is invari-
+ant under ordering, the second is using the defi-
+nition of fidelity and that π⊗n
+d
+has uniform coeffi-
+cients, and the final equality is the definition of the
+(k, p)−norms. In particular note we have dropped
+the sorting.
+We remark we could have set |φ⟩ = |φ′⟩⊗m to get
+a tradeoff, but this does not seem to provide any
+insight.
+Just as in the one-shot setting, we know the above
+result isn’t as useful in general because it can’t
+throw out resources, so we now present the general
+result.
+Proposition 12. The optimal fidelity of converting
+n d−local dimensional EPR pairs to |ψ⟩ under local
+operations is given by
+FLO(|ψ⟩ ,
+��Φ+
+d
+�⊗n) = d−n
+max
+P′∈P(Σ) ∥(P ⊗ P′)∥(dn,1/2) ,
+where ∥ · ∥(k,p) is (k, p)−norm generalized to p ≥ 0.
+Moreover, for fixed n, this is a convex optimization
+problem.
+
+17
+Proof. Starting from the result of Theorem 6,
+max
+P′∈P(Σ) F((P ⊗ P′)↓, (π⊗n
+d )↓)
+= max
+P′∈P(Σ) F((P ⊗ P′)↓, π⊗n
+d )
+=
+�
+�
+1
+dn/2
+max
+P′∈P(Σ) ∑
+i∈[dn]
+�
+(P ⊗ P′)↓(i)
+�
+�
+2
+(⋆)
+= 1
+dn
+max
+P′∈P(Σ) ∥P ⊗ P′∥(dn,1/2) ,
+the first inequality is invariance of π⊗n
+d
+under sort-
+ing, the second is definition of fidelity and that each
+element of π⊗n
+d
+is the same, the last is the definition
+of (k, p)-norm extended to p ≥ 0.
+To show this is a convex optimization problem,
+note that ΦP(·) := P ⊗ · is linear, −√· is opera-
+tor convex, and the sum of the k largest eigenvalues
+of a PSD P, which we will denote Σk(P) is convex.
+Thus, starting from (⋆),
+�
+�d−n/2
+max
+P′∈P(Σ) ∑
+i∈[dn]
+√
+P ⊗ P′↓(i)
+�
+�
+2
+=
+�
+−d−n/2
+min
+P′∈P(Σ)Σdn
+�
+−
+�
+ΦP(P′)
+��2
+,
+where
+we
+have
+used
+maxx∈C f (x)
+=
+− minx∈C − f (x) and our definitions.
+Then ig-
+noring the −d−n/2 factor and the square, the
+optimization
+problem
+is
+over
+the
+probability
+simplex, which is a convex subset of the positive
+semidefinite matrices, and the objective function
+is convex over the positive semidefinite cone as
+−
+�
+ΦP(·) is operator convex and Σdn is a convex
+function over the space of Hermitian operators.
+this completes the proof.
+Unfortunately,
+while
+this
+gives
+computable
+bounds, it is not clear how one could determine the
+optimal value analytically.
+2.
+Distillation Under Local Operations
+We now present the same results in the distilla-
+tion case, where we take some state to many EPR
+states.
+For completeness, we state the local uni-
+taries case.
+Proposition 13. The fidelity of distillation under lo-
+cal unitaries and zero communication is given by
+FLU(
+��Φ+
+d
+�⊗m , |ψ⟩⊗n) = d−m∥P⊗n∥|S|,1/2 ,
+where S = [min{dm, rank(P)n}].
+Proof. The proof is effectively identical to the dilu-
+tion case by symmetry of the fidelity.
+In contrast to the local unitary case, the symmetry
+is broken when one considers local operations.
+Theorem 10. For fixed d, m, n the optimal fidelity
+for dilution under local operations is given by
+FLO(
+��Φ+
+d
+�⊗m , |ψ⟩⊗n)
+=d−m
+�
+min
+P′∈P↓(Σ) − ∑
+i∈I
+αi
+�
+p′(i)
+�2
+,
+where P↓(Σ) is the set of decreasing distributions
+as defined in Section III, I ≡ [⌈rank(P)n/dm⌉], and
+αi := ∑j∈[(i−1)dm:min{i·dm,rank(P)n}]
+�
+p↓
+n(i). Note the
+minimization is a convex optimization program.
+Proof. Yet again, we use the square root fidelity and
+then take the square at the end. Then, using Theo-
+rem 6, we have
+FLO((Φ+
+d )⊗m, ψ⊗n)
+= max
+P′∈P(Σ) F((π⊗m
+d
+⊗ P′)↓, (P⊗n)↓)
+=
+�
+max
+P′∈P(Σ) ∑
+i∈S
+�
+(π⊗m
+d
+⊗ p′)↓(i)
+�
+p↓
+n(i)
+�2
+.
+Next, note
+(π⊗m
+d
+⊗ P′)↓ = d−m/2 ∑
+i′∈Σ
+p↓(i′)1Cdm ,
+where we have just used that π⊗m
+d
+is invariant
+under ordering.
+It follows that if we let I
+≡
+[⌈rank(P)n/dm⌉], we can rewrite,
+FLO((Φ+
+d )⊗m, ψ⊗n)
+=d−m
+�
+max
+P′∈P(Σ) ∑
+i∈I
+�
+(p′)↓(i)
+·
+∑
+j∈[(i−1)dm:min{i·dm,rank(P)n}]
+�
+p↓
+n(i)
+�2
+.
+Now
+first
+define
+αi
+:=
+∑j∈[(i−1)dm:min{i·dm,rank(P)n}]
+�
+p↓
+n(i)
+as
+these
+co-
+efficients may be pre-computed.
+Second, note
+that the probability simplex restricted to de-
+scending distributions, P↓(Σ) is itself convex as
+r↓
+λ := λp↓ + (1 − λ)q↓ satisfies
+λp↓(i) + (1 − λ)q↓(i) ≥ λp↓(i + 1) + (1 − λ)q↓(i) ,
+
+18
+for all i ∈ [|r|]. Thus we have,
+FLO(
+��Φ+
+d
+�⊗m , |ψ⟩⊗n)
+=
+�
+− d−m
+min
+P′∈P↓(Σ) − ∑
+i∈I
+αi
+�
+p′(i)
+�2
+.
+The minimization is a convex optimization problem
+because if we consider f (p′) := − ∑i αi
+�
+p′(i), then
+its Hessian is ∇2 f = ∑i[αi/4p′(i)−3/2] |i⟩⟨i|, which
+is positive semidefinite on the interior of the prob-
+ability simplex (i.e. when p′(i) > 0 for all i). This
+completes the proof.
+3.
+Two Qubit Setting
+We have now seen that even in the basic dilu-
+tion and distillation setting, while we can deter-
+mine convex optimization programs, we can’t seem
+to get clean analytic results. In this section we con-
+sider an even more tractable setting to attempt to
+resolve this: many copy two-qubit seed and target
+states. We show in this setting under certain as-
+sumptions the local unitary strategy is optimal and
+lobby this to show in particular that the optimal fi-
+delity of converting n copies of |φ⟩ to n copies |ψ⟩
+goes to zero as n goes to infinity. We note that this
+setting is more manageable because we effectively
+only have to reason about Bernoulli distributions.
+Lemma 11. Given Bernoulli distribution P
+=
+p |0⟩⟨0| + (1 − p) |1⟩⟨1|, then P⊗n is such that the
+sequence xn with (n − k) zeros has probability
+pn−k(1 − p)k.
+Moreover, there are (n
+k) sequences
+with probability pk(1 − p)n−k and the same for
+pn−k(1 − p)k.
+Proof. The claim that xn with (n − k) zeros has prob-
+ability pn−k(1 − p)k is straightforward. The second
+point actually just follows from the fact there are (n
+k)
+sequences with k zeros, which could be proven by
+induction in a straightforward manner.
+We can now use the above lemma along with
+Theorem 5 to get the optimal LU fidelity as a func-
+tion of the number of copies n.
+Corollary 3. Consider entangled states |ψ⟩ , |φ⟩ ∈
+C2 ⊗ C2. Then,
+FLU(ψ⊗n, φ⊗n)
+= ∑
+k∈[n]
+�n
+k
+�
+(pq)(n−k)/2((1 − p)(1 − q))k/2 .
+Proof. By
+Theorem
+5
+we
+can
+reduce
+to
+the
+Bernoulli distributions from the Schmidt coeffi-
+cients, |ψ⟩⊗n �→ P⊗n, |φ⟩⊗n �→ Q⊗n. Since these are
+Bernoulli distributions, if we assume without loss
+of generality p ≥ (1 − p), we can order the proba-
+bilities simply by the exponent, e.g. pj−k(1 − p)k ≥
+pj−k−k′(1 − p)k+k′ for any 0 ≤ k′ ≤ j − k. More-
+over, the cardinality of each set of sequences will be
+the same for both P⊗n and Q⊗n because |ψ⟩ , |φ⟩ are
+only entangled if their Schmidt rank is two. There-
+fore,
+F((P⊗n)↓, (Q⊗n)↓)
+= ∑
+k∈[n]
+�n
+k
+�
+(pq)(n−k)/2((1 − p)(1 − q))k/2
+where the sum is over the number of zeros in the
+string, the cardinality was proven in the previous
+lemma, and the last term is just a re-writing of
+�
+pn−k(1 − p)k
+�
+qn−k(1 − q)k.
+We note it is straightforward to generalize the
+above result to the case where you have the num-
+ber of states differs between the seed and the target,
+but the form would be ugly as one would need to
+count how many sequences of a given probability
+there are and keep track of this in the sum. Indeed
+at this point the problem is elaborate enough that
+there is no advantage with dealing with two-qubit
+states as it’s a question of the type classes [24]. We
+state this as a remark.
+Remark 1. Consider states |ψ⟩ , |φ⟩ respectively
+with ordered probability distributions correspond-
+ing to their Schmidt coefficients, P and Q respec-
+tively. FLU(|ψ⟩⊗n , |φ⟩⊗m) can be computed. This is
+because the probability of a given sequence drawn
+in i.i.d. form from a distribution has a closed form
+[24, Theorem 11.1.2]. It follows that as long as one
+determines the type classes exactly and takes into
+account that the sizes of the type classes may dif-
+fer between P and Q, the computation is possible,
+albeit tedious.
+Rather than dealing with the computational
+nightmare of generalizing beyond two qubit states,
+we now show that the term in Corollary 3 always
+goes to zero as n goes to infinity.
+Proposition
+14.
+Consider
+entangled
+states
+|ψ⟩ , |φ⟩ ∈ C2 ⊗ C2.
+lim
+n→∞ FLU(|φ⟩⊗n , |ψ⟩⊗n) = 0 .
+Proof. Let the probability distributions correspond-
+ing to their Schmidt coefficients be parameterized
+
+19
+by p and q = p + ε where ε ∈ [−1/2, 1/2]. Then,
+That way,
+FLU(|ψ⟩⊗n , |φ⟩⊗n)
+= ∑
+k∈[n]
+�n
+k
+�
+(p2 + ε)(n−k)/2
+· [(1 − p)2 − ε(1 − p)]k/2 .
+Now note p2 + ε < 1 as otherwise |ψ⟩ is product.
+Define α := (p2 + ε)1/2 < 1. Then we have
+�n
+k
+�
+(p2 + ε)(n−k)/2 · [(1 − p)2 − ε(1 − p)]k/2
+≤
+�n · e
+k
+�k
+αn−k[(1 − p)2 − ε(1 − p)]k/2
+=
+� e
+k α−1�k
+[(1 − p)2 − ε(1 − p)]k/2nk · αn
+=O(poly(n))O(exp(−n))
+→0 ,
+where in the inequality we have used an upper
+bound on the binomial coefficient, in the first equal-
+ity we have grouped terms by scaling, in the next
+equality we have used that the first portion is a
+polynomial in n and that α < 1, so αn scales in-
+verse exponentially in n. The limiting factor is then
+because an inverse exponential times a polynomial
+goes to zero. We also remark that the term where
+k = n will also go to zero as [(1 − p)2 − ε(1 − p)]k/2
+will go to zero as k goes to infinity as its magnitude
+will be bounded by 1.
+Therefore, each term in the sum goes to zero as
+n goes to infinity, so the entire sum will go to zero.
+This completes the proof.
+We note our proof tells us nothing about the scal-
+ing as a function of the difference between p and q
+nor does it tell us how fast it goes to zero compared
+to F(P⊗n, Q⊗n). These are shown numerically for
+specific cases in Fig. 4.
+It is then natural to ask if what we have seen so
+far is something special to local unitaries. We show
+that under sufficient conditions, just like in the sin-
+gle copy case, when two-qubit seed states are in-
+volved, local unitary strategies are optimal.
+Theorem 12. Let |ψ⟩ ∈ C2 ⊗ C2 and the target state
+be |ψ⟩⊗n. Let the seed state |φ⟩ satisfy SR(|φ⟩) ≤
+nSR(|ψ⟩). Then the optimal local operations strat-
+egy is the optimal local unitary strategy.
+Proof. By Theorem 6,
+FLO(|ψ⟩⊗n , |φ⟩)
+(a) Fidelity under local unitaries as n grows for
+various choices of q = p + ε.
+(b) Fidelity under local unitaries as n grows for
+various choices of q = p + ε compared to F(P, Q)⊗n.
+FIG. 4: Degradation of fidelity of trying to convert n
+copies of one pure two-qubit entangle state to another
+for various differences in Schmidt coefficients, q = p + ε
+where we choose p = 0.55. (a) Shows the rate that the
+local unitary strategy degrades is a nonlinear function of
+the size of ε. (b) Compares to the case where one does
+not re-order the Schmidt coefficients, i.e. compares to
+F(P, Q)n.
+= max
+P′∈P(Σ) F((P⊗n ⊗ P′)↓, Q↓)
+= ∑
+i∈|Q|
+�
+Q↓(i)
+�
+(P⊗n ⊗ P′)↓(i) .
+We will show that P′ should be the delta distribu-
+tion. If p ̸= 1/2, p′(1) < 1, then for any 0 ≤ k ≤ n,
+we have the inequalities
+pn−k(1 − p)k >pn−k(1 − p)kp′(1)
+>pn−k(1 − p)kp′(2)
+and
+pn−k(1 − p)k >pn−k(1 − p)kp′(1)
+>pn−(k+1)(1 − p)k+1p′(1)
+>pn−(k+1)(1 − p)k+1p′(2)
+As square root is a monotone, this holds when
+we take the square root. Note that by assumption
+
+ConvergenceComparison
+1.0
+0.8
+UnorderedE=0.4
+OrderedE=0.4
+0.6
+idelity
+Unordered E=0.1
+0.4
+OrderedE=0.1
+UnorderedE=0.05
+0.2
+OrderedE=0.05
+0.0
+0
+50
+100
+150
+200
+250
+Numberof CopiesnConvergenceComparison
+1.0
+0.8
+UnorderedE=0.4
+OrderedE=0.4
+0.6
+idelity
+Unordered E=0.1
+0.4
+OrderedE=0.1
+UnorderedE=0.05
+0.2
+OrderedE=0.05
+0.0
+0
+50
+100
+150
+200
+250
+Numberof Copiesn20
+P⊗n has enough entries by itself for there to be
+one corresponding to each q↓.
+Therefore, given
+the inequalities above, it follows if p′(1) ̸= 1, each
+term in the sum only decreases. Therefore, p′(1) is
+optimal for every n and k. Thus, when p ̸= 1/2,
+the optimal value is obtain by P′ being a delta
+distribution, which means it’s equivalent to the
+local unitary strategy.
+Finally, if p = 1/2, then pn−k(1 − p)k = 2−n for
+all k. Therefore, if p′(1) < 1, the inequalities simpli-
+fies for all 0 ≤ k ≤ n:
+pn−k(1 − p)kp′(1) =pn−(k+1)(1 − p)k+1p′(1)
+>pn−(k+1)(1 − p)k+1p′(2)
+and
+pn−k(1 − p)kp′(1) > pn−k(1 − p)kp′(2) .
+Again because each q term is paired up already, this
+means if p′(1) ̸= 1, the value decreases. Therefore,
+we again conclude the optimal strategy is the LU
+strategy. This completes the proof.
+We note that a trivial example of why we need
+the Schmidt rank constraint in the previous theo-
+rem is our original example for the advantage of
+LO strategies: if |φ⟩⊗n+ℓ where ℓ ≥ 1, then there
+is a better LO strategy than an LU strategy. Finally,
+we note it immediately follows from these previous
+results that
+Corollary 4. If |φ⟩ , |ψ⟩ ∈ C2 ⊗ C2 are both entan-
+gled, then
+lim
+n→∞ FLO(|ψ⟩⊗n , |φ⟩⊗n) = 0 .
+VI.
+ON CATALYTIC CONVERSION
+We now have established a rather robust theory
+of pure state transformations under local opera-
+tions. It is natural to return to the topic of conver-
+sion of one state to another using an ancillary en-
+tanglement, i.e. cataltyic transformations, which is
+a special case of the setting, and includes quantum
+embezzlement. Of course, it is immediate from our
+results so far that we know the optimization pro-
+gram that determines the optimal pure state cata-
+lyst, as we state in the following proposition.
+Proposition 15. For any Schmidt rank d, the opti-
+mal pure state catalyst for state conversion |φ⟩ to
+|ψ⟩ is the quantum state |ζ⟩ = vec(
+√
+R) that is de-
+termined via the optimization
+max
+R∈P(d),P′∈P(Σ) F((P ⊗ P′)↓, (Q ⊗ R)↓) .
+Proof. This immediately follows from the input be-
+ing |φ⟩ ⊗ vec(
+√
+R) and then applying Theorem 6.
+Note this means |Σ| scales as function of d.
+However, as we have already addressed, even
+without a free variable for the catalyst, the opti-
+mization in Theorem 6 seems unmanageable di-
+rectly. While in principle one could use the relax-
+ation in Theorem 9 to obtain efficient upper bounds,
+it is less obvious how often these will be non-trivial
+given that R is a free variable.
+The next most natural setting would be that of
+catalytic state conversion under local unitaries, i.e.
+we consider transformations of the form
+|φ⟩ |ζ⟩
+LU
+←→ ≈ε |ψ⟩ |ζ⟩ ,
+where |ζ⟩ is the catalytic resource and the arrow
+going in both directions is because local unitaries
+are reversible. This may be seen as a generaliza-
+tion of embezzlement where |φ⟩ = |0⟩A |0⟩B and
+|ζ⟩ = |µ(n)⟩.2
+Now as noted in the background, embezzling is
+known to be in effect optimal for sufficiently small
+ε. It follows for sufficiently small error ε > 0, the
+strategy that embezzles out the seed state and then
+embezzles in the target state is roughly optimal, i.e.
+|φ⟩ |µ(n)⟩
+LU
+←→ |0⟩ |0⟩ |µ(n)⟩
+LU
+←→ |ψ⟩ |µ(n)⟩
+is effectively optimal. Nonetheless, we may explore
+at what point this becomes necessary.
+Using Theorem 5, we know the optimal strategy
+is given by3
+max
+R∈P(d) F((P ⊗ R)↓, (Q ⊗ R)↓) .
+Even in the case P, Q, R ∈ P(2) this technically
+can’t be solved using gradient methods as one has
+to sort the p(1 − r) and (1 − p)r terms of p ⊗ r and
+likewise for q ⊗ r. Nonetheless, it is hopefully clear
+that r ∈ [min{p, q}, max{p, q}], as it is trying to
+make the distributions be more similar. Nonethe-
+less, this issue will only grow in difficulty with the
+dimension and it is unclear how one would prove
+an ansatz is optimal in general. Therefore, we pro-
+vide two-qubit examples which characterizes the
+general insights.
+2 We refer the reader to Proposition 3 if the notation has been
+forgotten.
+3 We stress that by the correspondence of Schmidt coefficients to
+probability distributions as discussed at the start of the work,
+even without Theorem 5, this would be a legitimate strategy,
+we simply wouldn’t know analytically it was optimal.
+
+21
+Example 4 (Resource Gap Between Embezzling
+and Optimal Catalyst). Consider Bernoulli distri-
+butions P, Q, R parameterized by p = 0.5, q = 0.7
+and we leave r unspecified for now. In other words,
+one of the states is the maximally entangled states
+and the other is, up to local unitaries,
+√
+0.7 |00⟩ +
+√
+0.3 |11⟩. Therefore, depending on which way one
+runs the transformation, we are considering entan-
+glement dilution or distillation with a catalytic re-
+source. Without the resource,
+FLO(|ψ⟩ , |φ⟩) = F(P↓, Q↓) ≈ 0.958.
+One can verify that the optimal choice of r⋆ ≈ 0.6
+in this case. For this choice
+FLU(|ψ⟩ |ζ⟩ , |φ⟩ |ζ⟩) =F((P ⊗ R⋆)↓, (Q ⊗ R⋆)↓)
+>0.979 .
+The first problem is that 0.979 is not an accept-
+ably high fidelity even by contemporary standards.
+Nonetheless, note that to get this state via embez-
+zling (and ignoring that embezzling out the initial
+state introduces error), it would require generating
+|µ(n)⟩ where n > m1/(1−0.979) = 2 · 1014. That is,
+even to embezzle a two-qubit pure state would re-
+quire generating an inconceivable amount of entan-
+glement. For this reason, specially engineered cat-
+alysts seems a significant improvement up to any
+error that can be achieved.
+On the other hand, one might note that if we
+could generate R where r = 0.55, then we may as
+well have just used this state to begin with as
+F(P↓, R↓) =0.98989
+>F((P ⊗ R⋆)↓, (Q ⊗ R⋆)) .
+From a practical perspective we agree with this cri-
+tique. Nonetheless, from a basic science perspec-
+tive, if we are interested in local unitary conversions
+under catalysts, then the above tells us there are bet-
+ter choices in general than embezzlement, although
+embezzling has the special property of being uni-
+versal and optimal for sufficiently small ε.
+We close this consideration with two final re-
+marks. First, if one picks two states that are more
+similar to begin with, then the scaling of the embez-
+zling state will be even larger. Second, we have not
+presented how the fidelity for this example scales as
+the local dimension of |ζ⟩ grows. Both the dimen-
+sion scaling and two states that are more similar are
+considered in Fig. 5 where the near-optimal fideli-
+ties are found via brute force numerical search.
+(a) Maximum achievable fidelity of transformation
+under local unitaries as a function of the Schmidt
+rank of the catalyst.
+(b) Order of the Schmidt Rank of embezzling state
+|µ(n)⟩ to achieve same fidelity.
+FIG. 5: Plots regarding dimension scaling in Example 4.
+(a) The achievable fidelity of converting one two-qubit
+entangled state to another parameterized by p and q
+under local unitaries using a catalyst with a given local
+dimension (equivalently, Schmidt rank) using brute
+force search. (b) The order (i.e. the power of 10) in the
+Schmidt rank of the embezzling state |µ(n)⟩ to obtain
+the same maximum fidelity. This is calculated using
+21/(1−Fmax) following Proposition 3. All optimizer
+catalysts provided in an appendix for verification.
+VII.
+ON EXTENSIONS OF THE THEORY
+As a final consideration, we discuss the applica-
+tion of our results beyond bipartite pure states. First
+we remark upon extensions to multipartite pure
+states. In this case the problem is that in establish-
+ing all of the results, we have used that local uni-
+taries can take the Schmidt decomposition of the
+state to one of a canonical form. However, in the
+multipartite case, the Schmidt decomposition does
+not even exist in general [25]. As such this argu-
+ment immediately breaks down. Furthermore, in
+the proof of Theorem 5 we used Uhlmann’s theo-
+rem, which requires partitioning the state into two
+pieces, one of which is the purification. Therefore, it
+
+MaximumFidelityasaFunctionofCatalystDimension
+1.00
+0.99
+L
+lity
+Fideli
+0.98
+p=0.6,q=0.65
+p=0.5,q=0.7
+Max
+0.97
+0.96
+0
+2
+4
+6
+8
+AllowedCatalvstSchmidtRankOrderofEmbezzlingStateDimforsameMaxFidelity
+Schmidt Rank
+500
+100
+p=0.6,q=0.65
+OrderofEmbezzling
+50
+p=0.5,q=0.7
+10
+0
+2
+4
+6
+8
+AllowedCatalvstSchmidtRank22
+seems no multipartite extension of this work holds.
+Similarly,
+there are issues with approaching
+mixed states. One issue is to note that all relation-
+ships we have been able to establish have stemmed
+from the fidelity under local unitaries of pure states.
+Even in the case where local operations made a
+pure state no longer pure, we purified operations so
+that the states were pure. We simply cannot do this
+if we start with mixed states in both arguments of
+the fidelity. We also cannot purify the states as by
+data-processing, any optimization without tracing
+off the purifying space only gets us a lower bound.
+Moreover, this lower bound would require estab-
+lishing results for tripartite systems, which returns
+to the issues with the multipartite pure state case.
+Therefore, we believe in effect these are the most
+general settings where these proof methods will be
+of use.
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+Appendix A: Randomness Embezzling Proof and
+Discussion on Locality
+In this section we provide the proof of Theorem
+2 and then briefly discuss how it differs from quan-
+tum embezzlement.
+Proof. The proof is largely the same as for embezzle-
+ment of quantum states [12]. Let P = ∑i p(i) |i⟩⟨i|.
+Define Wn as Rn ⊗ P except with probabilities in de-
+
+23
+creasing order. Note
+Rn ⊗ P =
+1
+Hn ∑
+i,j
+p(i)
+j
+|i⟩⟨i| ⊗ |j⟩⟨j| ,
+so there exists a relabeling on {(i, j)} that will take
+this to Wn. In particular, letting f : [m] × [n] → [m ·
+n] be a bijection, we have |i⟩ |j⟩ → | f (i, j)⟩ ≡ |i′⟩ |j′⟩
+such that
+�
+z f (i,j) := p(i)
+jHn
+�
+(i,j) satisfy zk ≥ zk+1 for all
+k ∈ [m · n]. Therefore it suffices to approximate Wn,
+which means we want to bound the overlap of this
+with Rn ⊗ P.
+For fixed t and i, we let
+Nt
+i :=
+����
+�
+(i, j) : p(i)
+jHn
+>
+1
+tHn
+����� .
+The inequality may be manipulated to imply 1 ≤
+j < p(i)t. It follows that Nt
+i < p(i)t. From this we
+obtain ∑m
+i=1 Nt
+i < ∑m
+i=1 p(i)t < t, where we have
+used ∑i p(i) = 1.
+As z1 ≥ z2 ≥ ..., it follows
+zj ≤
+1
+jHn for all 1 ≥ j ≥ n. We may restate this as for
+1 ≤ j ≤ n, there are at most t′ − 1 pairs (i, j) such
+that p(i)/(jHn) > 1/(t′Hn). Recalling z1 ≥ z2 ≥ ...,
+this means that z1 < 1/Hn and that there is at most
+one pair (i, j) pair such that p(i)/(jHn) < 1/(2Hn),
+which, since z1 ≥ z2, means if such a pair exists,
+it is z1. By applying this argument in effect recur-
+sively, we see that for t′, there are at most t′ − 1
+(i, j) pairs such that p(i)/(jHn) > 1/(t′Hn) and
+since zk ≥ zk+1, if all of these pairs exist, then it
+must be z1, ..., zt′−1. Therefore, zj ≤ 1/(jHn) for all
+1 ≤ j ≤ n. We can now use this to bound the fi-
+delity.
+F(Rn ⊗ |0⟩⟨0| , Wn) =
+�
+n
+∑
+j=1
+�
+zj
+jHn
+�2
+≥
+�
+n
+∑
+j=1
+�zj
+�2
+≥
+n
+∑
+j=1
+zj,
+where in the equality we have used the definition
+of fidelity, in the second we used our established in-
+equality, and in the third we have used √x + √y ≥
+√x + y for x, y ≥ 0 to pull the square root out
+around the sum and cancel with the square.
+Now we want to lower bound this sum, which
+requires managing the zj terms.
+We consider
+Tn = Rn ⊗ πm with probabilities t(j) where πm :=
+1
+m ∑m
+i=1 |i⟩⟨i|.
+Now note that zk ≥ tk for all k ∈
+[m · n], and this is independent of what the distri-
+bution P is. We can then bound the relevant sum by
+the sum for Tn. It follows
+n
+∑
+j=1
+tj =
+⌊n/m⌋
+∑
+j=1
+m
+∑
+i=1
+1
+jHnm =
+⌊n/m⌋
+∑
+j=1
+1
+jHn
+=
+H⌊n/m⌋
+Hm
+≥ln(n/m)
+ln(n)
+= 1 − log(m)
+log(n) ,
+where the second inequality is using Hn ≥ ln(n)
+and the final form is converting from ln to log in
+both the numerator and denominator so it cancels.
+Finally, leting 1 − log(m)/ log(n) > 1 − ε will result
+in n > m1/ε, which completes the proof.
+With the proof established, we expand upon the
+distinction between the entangled and classical dis-
+tribution cases of embezzlement in terms of local-
+ity briefly mentioned in the main text. In the clas-
+sical case, one party embezzles a distribution lo-
+cally by themselves, whereas in the entangled case
+two parties act locally on a non-local distribution.
+Mathematically, this simply follows from the fact
+the vec(·) map and its inverse converts between bi-
+partite states and a probability distribution. How-
+ever, it is also physically interesting that these are
+the two cases that align as it is clear other varia-
+tions are either classically or quantumly impossible
+as we now explain.
+The first reasonable variation would be if there
+is a non-local classical case where two parties try
+and construct some joint distribution pXY using cat-
+alyst rX′Y′. It is easy to see that they cannot in gen-
+eral satisfy the decoupling condition that is satisfied
+in quantum embezzlement, i.e. they cannot satisfy
+pXY ⊗ rX′Y′ in this setting. This is because without
+loss of generality the state will be of the form
+qXYX′Y′ = ∑
+x,x′,y,y′
+q(x|x′)q′(y|y′)r(x, y)
+·
+��x, y, x′, y′��
+x, y, x′, y′�� .
+This form means that X will be correlated to X′ and
+Y to Y′ unless qXY may be generated non-locally
+without a seed state to correlate the two which
+means they are (up to the allowed error) indepen-
+dent, i.e. qXY ≈ε qX ⊗ qY. In this sense, there cannot
+be a classical non-local equivalent of quantum em-
+bezzlement.
+On the other hand, if one does not require the
+decoupling, then this is a task that is possible in
+the classical setting and is known as distributed
+source simulation, where the question is the min-
+imal needed shared randomness as the seed state to
+generate the target state up to an (arbitrary) er-
+ror [26].
+This was determined asymptotically in
+the classical case by Wyner [8], extended to sepa-
+rable states by Hayashi [9], and recently general-
+ized to the one-shot setting for separable states in
+[10]. However, as in this setting variation there is no
+
+24
+communication between the acting parties and the
+catalyst acts as the seed state, it follows from Propo-
+sition 2 that distributed source simulation cannot
+admit an entangled state equivalent. For these rea-
+sons, not only does the vec bijection specify the cor-
+respondence of embezzlement in the classical and
+quantum setting, but deviating from it makes either
+a quantum or classical version impossible.
+Appendix B: Semidefinite Program Relaxation of Max
+Fidelity of Pure State Transformation Under LOSR
+In this section we prove Theorem 9. We begin by
+establishing (7) is true.
+Lemma 13. Consider target state |ψ⟩ and seed state
+|φ⟩. Let SR(ψ) = d and SR(φ) = d′. Define A = Cd,
+B = Cd·d′. Then,
+FLOSR(|ψ⟩ , |φ⟩) ≤ max F(R, Q↓
+embed)
+s.t. TrB[R] = P↓
+R ∈ P↓(d2 · d′) ,
+where P and Q are the distributions defined by |ψ⟩
+and |φ⟩’s Schmidt coefficients respectively.
+Proof. The above seems intuitively true from The-
+orem 6 as we have just relaxed the tensor product
+structure with the partial trace constraint. The tech-
+nical issue is the ordering operation ·↓ is defined in
+terms of a permutation of a fixed basis, so we need
+to make sure this works with the partial trace.
+Note the feasible set, the set we can optimizer
+over, in Theorem 6 is S1(P) := {(P ⊗ P′)↓ : P′ ∈
+P(Σ)}. Now note this is the same as the set
+S2(P) := {(P↓ ⊗ P′↓)↓ : P′ ∈ P(Σ)} ,
+because the ordering applied to the tensor product
+will result in the same thing regardless of whether
+or not P, P′ were ordered. Therefore, we can focus
+on P↓ ⊗ P′↓ to make the explanation clearer.
+In general, in terms of vectors,
+(p↓ ⊗ p′↓)↓ =
+�
+�
+�
+�
+�
+�
+p↓(1)p′↓
+p↓(2)p′↓
+...
+p↓(d)p′↓
+�
+�
+�
+�
+�
+�
+,
+where p(i) ≥ p(i + k) for k ≥ 0. Formally, we also
+have
+p↓(i)p′↓(1) ≥ p↓(i + k)p′↓(j)
+for all i ∈ [d], k ∈ {0, ..., d − i}, and j ∈ Σ. In partic-
+ular what this means is that without loss of general-
+ity for any i ∈ [d], p↓(i)p′↓(1) appears before any el-
+ement that is not of the form p↓(i − ℓ)p′↓(j) for some
+0 < ℓ ≤ i − 1. It follows that under the ordering of
+(p↓ ⊗ p′↓)↓, when the partial trace marginalizes to
+the A space, the induced ordering on the local space
+will be the ordering based on p↓. Formally, this can
+be expressed as
+TrC|Σ|[(P↓ ⊗ P′↓)↓]
+= ∑
+j∈Σ
+1A ⊗ ⟨j| (P↓ ⊗ P′↓)↓ |j⟩
+= ∑
+i∈[d]
+p↓(i) |i⟩⟨i| ,
+where the first equality is a representation of the
+partial trace and the second is using the property
+noted of the ordering on the joint ordered distribu-
+tion.
+Thus, if X ∈ S2(P), TrC|Σ|(X) = P↓ and X ∈
+P↓(d · |Σ|). Noting that |Σ| = d · d′, this is the fea-
+sible set we have defined in the proposition. This
+completes the proof.
+The remaining point is to prove this is the
+semidefinite program given in (8). There is much to
+the theory of semidefinite programs for quantum
+information [19], but for our purposes all we will
+need is the following definition.
+Definition 2. A semidefinite program may be ex-
+pressed as
+max Tr(AX)
+s.t. Φ(X) = B
+XCd ⪯ 0 ,
+where Φ ∈ T(Cd, Cd′) is a Hermitian-preserving
+map, A ∈ Herm(Cd), B ∈ Herm(Cd′), and Herm(·)
+is the space of Hermitian operators on a given
+Hilbert space.
+The fidelity is known to be a semidefinite pro-
+gram [19], so we are really just verifying all of our
+constraints work and that we can write the SDP
+simply by making use of that.
+Lemma 14. The optimization program in the pre-
+vious lemma, may be expressed as the following
+
+25
+semidefinite program over the reals.
+max ∑
+i∈[d2·d′]
+x(i)
+s.t.
+�diag(r)
+diag(x)
+diag(x) diag(q↓
+embed)
+�
+⪰ 0
+TrB[diag(r)] = P↓
+r ∈ P↓([d2 · d])
+x ∈ Rd2·d′ ,
+where d, d′ are defined in the previous lemma.
+Proof. We begin by expressing the objective func-
+tion of the previous lemma, which is in terms of
+fidelity, using the primal problem for the SDP for
+fidelity from [19, Theorem 3.17]:
+max1
+2
+�
+Tr(X) + Tr
+�
+X†��
+� R
+X
+X† Q↓
+embed
+�
+≥ 0
+X ∈ L(C[d2·d′]) .
+Now our goal is to reduce X to the diagonal of a
+real vector.
+Note that R, Q↓
+embed are always invariant under
+pinching onto the computational basis of C[d2·d′],
+which we can denote ∆. Note that this pinching is
+a CPTP, so by the CP property,
+(idC2 ⊗ ∆)
+� R
+X
+X† Q↓
+embed
+�
+=
+�
+R
+∆(X)
+∆(X†) Q↓
+embed
+�
+≥ 0 .
+It also then follows as a positive semidefinite oper-
+ator is always Hermitian that
+�
+R
+∆(X†)
+∆(X) Q↓
+embed
+�
+≥ 0 .
+Thus by taking these two cases and averaging them,
+we have that
+�
+R
+1
+2
+�
+∆(X + X†)
+�
+1
+2
+�
+∆(X + X†)
+�
+Q↓
+embed
+�
+≥ 0 .
+Define X := 1
+2
+�
+∆(X + X†)
+�
+. Then note
+1
+2
+�
+Tr(X) + Tr
+�
+X†��
+=1
+2
+�
+Tr(∆(X)) + Tr
+�
+∆(X†)
+��
+=1
+2
+�
+Tr
+�
+X
+� + Tr
+�
+X†��
+= Tr
+�
+X
+�
+,
+where the first equality is because the pinching is
+trace preserving, the second is by definition of X,
+as is the final equality. Thus, for any X that sat-
+isfies the positivity constraint, we could replace it
+with X without loss of generality as we are con-
+sidering a maximization. Finally, note that X is a
+real diagonal matrix by the pinching along with the
+fact a + a∗ = 2 Re{a}. Thus X = diag(x) for some
+x ∈ Rd2·d′. Combining all these points and using
+Tr
+�
+X
+� = ∑i∈[d2·d′] x(i), we have reduced to consid-
+ering
+max ∑
+i∈[d2·d′]
+x(i)
+�diag(r)
+diag(x)
+diag(x) diag(q↓
+embed)
+�
+≥ 0
+x ∈ Rd2·d′ .
+This argument works for any choice of diagonal r,
+so this is the major reduction.
+What remains is to prove all the constraints are
+Hermitian maps. One can write the constraints for
+r ∈ P↓ as r(i) ≥ r(i + 1) for all i, which are semidef-
+inite constraints and can be written as Hermitian
+preserving maps on the variables r, x.
+diag is a
+Hermitian preserving map as is the partial trace, so
+TrC[diag(r)] is a Hermitian preserving map. Like-
+wise is the block matrix mapping if one allows for
+the complex conjugate in the lower left block, but
+noting diag(x)† = diag(x), we can leave it as writ-
+ten. Thus all the maps are Hermitian-preserving.
+The conversion to actual standard form we then
+omit as it provides no insight. This completes the
+proof.
+The above two proofs establish Theorem 9.
+Appendix C: Data for Catalyst Figure
+For p = 0.5, q = 0.7:
+Dimension Optimal distribution r
+1 n/a
+2
+1
+100[4, 6]
+3
+1
+100[21, 32, 47]
+4
+1
+100[12, 18.5, 0.28, 0.415]
+5
+1
+100[7, 11, 17, 26, 39]
+6
+1
+100[5, 8, 11, 16, 24, 35]
+7
+1
+100[3, 5, 8, 11, 16, 23, 24]
+8
+1
+100[5, 6, 9, 9, 13, 14, 19, 25]
+
+26
+For p = 0.6, q = 0.65:
+Dimension Optimal distribution r
+1 n/a
+2
+1
+100[32, 63]
+3
+1
+100[18, 31, 51]
+4
+1
+100[10, 17, 28, 45]
+5
+1
+100[6, 10, 16, 26, 42]
+6
+1
+100[7, 11, 12, 18, 20, 32]
+7
+1
+100[0, 7, 11, 12, 18, 20, 32]
+8
+1
+100[0, 0, 7, 11, 12, 18, 20, 32]
+