Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture

I show that if a finite-dimensional density matrix has strictly smaller von Neumann entropy than a second one of the same dimension (and the rank is not bigger), then sufficiently (but finitely) many tensor-copies of the first density matrix majorize a density matrix whose single-body marginals are all exactly equal to the second density matrix. This implies an affirmative solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC transfer to the classical setting of finite-dimensional probability vectors (with permutations of entries instead of unitary transformations for the CEC).


(top):
Conjecture 1 (CEC). Let ρ, ρ be d-dimensional density matrices that are not unitarily equivalent. Then there exists a finite-dimensional density matrix σ and a unitary U such that if and only if H(ρ) < H(ρ ) and rank(ρ) ≤ rank(ρ ).
An affirmative solution to the CEC provides a clean single-shot interpretation of von Neumann entropy without external randomness using the "catalyst" σ, see Refs. [1,2] for discussions of the conjecture and their physical implications. In Ref. [2], an approximate version of the CEC was proven. It follows directly from the proof, which itself is similar to earlier constructions in Refs. [3][4][5], that the CEC is true if the following Lemma holds true, illustrated in Fig. 1 (bottom). For completeness, we provide the explicit construction of U and σ in the CEC based on the following Lemma in the appendix. operation U is applied to systems S and C in the state ρ ⊗ σ. The resulting reduced state on S is exactly ρ , while the reduced state on C is preserved exactly, but correlated to S (indicated by the dashed lines). Both σ and U may depend on ρ and ρ . Bottom (Lemma 1): For some finite n, the state ρ ⊗n majorizes a state ρ n with all marginals equal to ρ . The results of this paper show that if ρ and ρ are not unitarily equivalent, then both situations occur if and only if H(ρ) < H(ρ ) and rank(ρ) ≤ rank(ρ ). Both results transfer to the classical setting as well. Lemma 1. Let ρ, ρ be d-dimensional density matrices that are not unitarily equivalent. Then there exist n ∈ N and a density matrix ρ n on n copies of C d such that ρ ⊗n ρ n , Tr {1,...,n}\i [ρ n ] = ρ ∀i = 1, . . . , n (2) if and only if H(ρ) < H(ρ ) and rank(ρ) ≤ rank(ρ ).
Here, for any two density matrices ρ and σ, ρ σ ("ρ majorizes σ") means that there exists a finite collection of unitaries U i with associated probabilities p i It acts on any operator X as D σ [X] = j |j j | X |j j |, but can be represented as i for a suitable set of d unitaries U i . An equivalent definition of majorization in terms eigenvalues is as follows (by the Schur-Horn Lemma and Birkhoff's Theorem, see for example Refs. [6,7]): if λ λ λ ρ denotes the vector of eigenvalues of ρ (including multiplicities) and λ λ λ σ the corresponding vector of eigenvalues of σ, then there exists a finite collection of permutation matrices Π i and a probability distribution q i such that λ λ λ σ = i q i Π i λ λ λ ρ . This latter characterization of majorization is also the appropriate one for general probability vectors.
In words Lemma 1 reads: if the entropy of ρ is higher than that of ρ (and the rank not smaller), then for some finite n the state ρ ⊗n majorizes a state whose one-body marginals are all equal to ρ . It is well known that for sufficiently large n the state ρ ⊗n majorizes an -approximation of ρ ⊗n . Lemma 1 instead shows that ρ ⊗n majorizes a state whose marginals coincide exactly with ρ , but where arbitrary correlations are allowed. Our aim here is to prove Lemma 1 and thereby the exact catalytic entropy conjecture. In fact, the arguments below can also all be transferred to the classical setting, where density matrices are replaced by finite-dimensional probability vectors and unitary transformations by permutations of the entries of the vectors. Using the corresponding classical construction in Ref. [2] this implies that the classical version of the CEC also holds (contrarily to some prior beliefs, see Supplemental Material of Ref. [1]): Theorem 2 (classical CEC). Let p p p, p p p ∈ R d be probability vectors that are not related by a permutation of their entries. Then there exists a finite-dimensional probability vector q q q and a permutation matrix Π such that In the formulation of Theorem 2, Tr x denotes marginalization over tensor-factor x and rank(x x x) denotes the number of non-zero elements of a vector x x x.
In fact, contrary to the usual situation where results in quantum theory imply corresponding classical results, Theorem 2 implies the quantum version of the CEC but not vice-versa. However, as in the approximate case, the dimension required for the catalyst can be significantly smaller in the quantum case compared to the classical case, see Ref. [2].

Resource theories
In Refs. [5,[8][9][10] catalysts that become correlated to the system of interest have been used to provide single-shot interpretations of the non-equilibrium free energy in quantum thermodynamics and the entanglement entropy in the framework of local operations and classical communication. These results were approximate, in the sense that the output state of a protocol only approximated the desired state up to arbitrary accuracy as the catalyst dimension diverged. Our result may suggest that a strict decrease in the free energy or entanglement entropy could imply the possibility of an exact transformation with a finite-dimensional catalyst also in those settings. However, Ref. [11] showed already that this is not the case, at least if one demands that the final correlations between catalyst and system can be made arbitrarily small. In the setting of this note, we cannot make the latter assumption since the mutual information between catalyst and final state of the system is always given by H(ρ ) − H(ρ). It would be interesting to know whether in the context of quantum thermodynamics a strict decrease in free energy allows for an exact conversion with finite-dimensional catalyst if one does not put a condition on the final correlations. Indeed, Ref. [12] showed that this is true in the quasi-classical case with the additional assumption that the thermal state only has rational entries in its eigenbasis.

Open problems
Lemma 5 does not claim that the state ρ n is close to ρ ⊗n .
It is an open problem whether one can both arrange that the marginals of ρ n are exactly equal to ρ and ρ n is arbitrarily close to ρ ⊗n in trace-distance while keeping the majorization condition ρ ⊗n ρ n . If true, this statement could likely have many further applications in physics and information theory.
A second open problem concerns the case H(ρ) = H(ρ ). It is known from lower bounds on the catalyst dimension that in this case a finite-dimensional catalyst in general cannot be used to implement the state-transition exactly [13,14]. On the other hand, continuity of von Neumann entropy implies that it can be implemented arbitrarily well as the catalyst dimension diverges. However, it is presently unknown whether a catalyst with properly infinite dimensional density matrix could be used to implement the state-transition ρ → ρ exactly in this case.

The proof
We formulate the proof in the quantum language, but, as mentioned above, everything transfers to the classical case by replacing density matrices with probability vectors. Since randomly permuting subsystems is a random unitary process (in the sense of majorization), whenever ρ ∈ S n (ρ) there exists a state ρ n with ρ ⊗n ρ n as in Lemma 1. So we want to show that whenever H(ρ ) > H(ρ) and rank(ρ) ≤ rank(ρ ) then ρ ∈ S n (ρ) for some n ∈ N . Proof. The set of density matrices σ such that ρ ⊗n σ is closed and convex and hence compact. S n (ρ) is convex and compact, because it is its image under a linear (and hence continuous) map. It has full dimension since in particular it includes an open ball around 1/d (and hence also 1/d).
We now introduce a form of typicality. Note that the statement is not in the usual form, as we claim a statement for all density matrices with certain entropies. In other words, we require an error bound that is uniform over all density matrices with a certain minimal entropy. To state the Lemma, we define the distance d(ρ, S) between a density matrix ρ and a set of density matrices S as d(ρ, S) = inf σ∈S d(ρ, σ), where d(ρ, σ) = 1 2 ρ − σ 1 denotes the trace-distance.
Lemma 5 (Typicality). For every δ > 0 and > 0 there exists n ∈ N such that for all states ρ with Proof. In general, a density matrix ρ majorizes an - where H max is the smooth max-entropy and H min the smooth min-entropy, see [15,16]. By Theorem 1 of Ref. [17] (in Ref. [17], the bounds are stated for conditional entropies of classical random variables, but in the unconditional case, they generalize immediately to the quantum case), it follows that for every δ ≥ 0 we have for = 2 − nδ 2 2 log 2 (d+3) (which does not depend on ρ or ρ) 1 . Thus, if we take we find that ρ ⊗n majorizes an -approximation ρ ,n of ρ ⊗n as long as H(ρ ) > H(ρ) + 2δ. By the triangle inequality and monotonicity of the trace-distance under quantum channels, we find: Tr {1,...,n }\i [ρ ,n ], ρ ) ≤ .
The crucial additional ingredient to our proof apart from the typicality Lemma is the following simple observation about convex geometry.
denote an open ball of radius δ around y and let C be a parametrized family of closed and convex sets with the following property: where d is the euclidean distance on R d . Then y ∈ C for all < δ/2. 1 We define the smoothing of the entropies in terms of tracedistance instead of 1-norm, which accounts for a factor of 1/2. Proof. Let < δ/2 and suppose contrarily that y / ∈ C . Let c be the element in C closest to y (which is unique because of convexity of C ). But since y is the center of the ball, there must then be an element z ∈ B y (δ) with ≥ d(z, C ) ≥ d(z, y) > δ/2, which is a contradiction. To find z, shoot a ray through y and c. The part of the ray inside of B y (δ) has length at least δ, so in particular there must be a corresponding element z with d(z, c) > δ/2. Corollary 7. Let C be a full-dimensional convex set in a normed, finite-dimensional, real vector space V and let C ⊆ V be a parametrized family of closed and convex sets with the following property: where d is the distance induced by the norm. Then for any y ∈ int C there exists an 0 such that y ∈ C for all < 0 .
Proof. Since y is an interior point, there is an open ball B y (δ) around it and we can apply Lemma 6 (after embedding everything in some R d ).
We can now prove Lemma 1: We start with the "if" part. By unitary freedom and because rank(ρ) ≤ rank(ρ ), we can assume that supp(ρ) ⊆ supp(ρ ). Indeed, the rankcondition simply means that the dimension of the support of ρ is not larger than the dimension of the support of ρ . Furthermore, we can shrink Hilbert-space to the support of ρ , so that ρ has full rank (ρ > 0). We will do so in the following. So let H(ρ ) − H(ρ) = 2δ > 0 and ρ > 0. Then ρ is an interior point of the convex set of all density matrices since it has full rank. Moreover, it has entropy H(ρ ) > H(ρ) + δ and von Neumann entropy is continuous. Therefore, there is an open ball of states around it (within the set of density matrices) all of which have entropy > H(ρ) + δ. Hence it is an interior point of the convex set C of density matrices with entropy ≥ H(ρ) + δ and C is full-dimensional. For any > 0 set C = S n (ρ) with n from Lemma 5. Then by the same Lemma, the conditions of Corollary 7 are fulfilled if we set y = ρ . Hence there exists some n such that ρ ∈ S n (ρ).
For the "only if" part let us start with the condition on the von Neumann entropy: We use that H is sub-additive, H(ρ 12 ) ≤ H(ρ 1 ) + H(ρ 2 ) with equality if and only if ρ 12 = ρ 1 ⊗ ρ 2 , and strictly Schur-concave, meaning that if ρ ρ but ρ and ρ are not unitarily equivalent, then H(ρ) < H(ρ ). So suppose that ρ ⊗n ρ n and ρ and ρ are not unitarily equivalent. Then we have We now assume H(ρ) = H(ρ ) to arrive at a contradiction. Then (12) implies H(ρ n ) = nH(ρ ), which in turn implies ρ n = ρ ⊗n . But then ρ ⊗n ρ ⊗n and H(ρ) = H(ρ ), which by strict Schur-concavity implies that ρ ⊗n and ρ ⊗n are unitarily equivalent. But then ρ and ρ are unitarily equivalent, which yields the contradiction.
Acknowledgements. I would like to thank Niklas Galke for rekindling my interest in this problem and pointing out a mistake in a previous version of the argument. I would also like to thank Paul Boes,