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arXiv:2402.18037 · Physical Review A

Nonpositive-transpose entanglement and bound entanglement: From distillability sets to inequalities and multivariable insights

Si-Yuan Qi, Geni Gupur, Yu-Chun Wu, Guo‐Ping Guo

Published Jul 1, 2024• 0 citations
arXivHTMLPDF

Abstract

Equivalence between positive partial transpose (PPT) entanglement and bound entanglement is a long-standing open problem in quantum information theory. Limited progress has been made so far, even on the seemingly simple case of the bound entanglement of Werner states. The primary challenge is to give a concise mathematical representation of undistillability. To this end, we propose a decomposition of the $N$-undistillability verification into $\text{log}(N)$ repeated steps of 1-undistillability verification. For Werner states' $N$-undistillability verification, a parameter interval for $N$-undistillability is given, which is independent of the dimensionality of Werner states. Equivalent forms of inequalities for both rank 1 and 2 matrices are presented, before transforming the 2-undistillability case into a matrix analysis problem. A new perspective is also attempted by seeing it as a nonconvex multivariable function, proving its critical points and conjecturing Hessian positivity, which would make them local minimums.

DOI
10.1103/physreva.110.012406

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Equivalence between positive partial transpose (PPT) entanglement and bound entanglement is a long-standing open problem in quantum information theory. Limited progress has been made so far, even on the seemingly simple case of the bound entanglement of Werner states. The primary challenge is to give a concise mathematical representation of undistillability. To this end, we propose a decomposition of the NNN-undistillability verification into log(N)\text{log}(N)log(N) repeated steps of 1-undistillability verification. For Werner states' NNN-undistillability verification, a parameter interval for NNN-undistillability is given, which is independent of the dimensionality of Werner states. Equivalent forms of inequalities for both rank 1 and 2 matrices are presented, before transforming the 2-undistillability case into a matrix analysis problem. A new perspective is also attempted by seeing it as a nonconvex multivariable function, proving its critical points and conjecturing Hessian positivity, which would make them local minimums.
An important open problem in quantum information theory is the question of the existence of NPT bound entanglement. In the past years, little progress has been made, mainly because of the lack of mathematical tools to address the problem. (i) In an attempt to overcome this, we show how the distillability problem can be reformulated as a special instance of the separability problem, for which a large number of tools and techniques are available. (ii) Building up to this we also show how the problem can be formulated as a Schmidt number problem. (iii) A numerical method for detecting distillability is presented and strong evidence is given that all 1-copy undistillable Werner states are also 4-copy undistillable. (iv) The same method is used to estimate the volume of distillable states, and the results suggest that bound entanglement is primarily a phenomenon found in low dimensional quantum systems. (v) Finally, a set of one parameter states is presented which we conjecture to exhibit all forms of distillability.
Abstract One of the oldest problems in quantum information theory is to study if there exists a state with negative partial transpose which is undistillable [1]. This problem has been open for almost 30 years, and still no one has been able to give a complete answer to it. This work presents a new strategy to try to solve this problem by translating the distillability condition on the family of Werner states into a problem of partial trace inequalities, this is the aim of our first main result. As a consequence, we obtain a new bound for the 2-distillability of Werner states, which does not depend on the dimension of the system. On the other hand, our second main result provides new partial trace inequalities for bipartite systems, connecting some of them also with the separability of Werner states. Throughout this work, we also present numerous partial trace inequalities, which are valid for many families of matrices.
Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bipartite partial-trace inequality for every matrix of rank at most two. As applications, we prove the two-copy undistillability of NPT Werner states in arbitrary local dimension, thereby resolving this open problem highlighted in [P. Horodecki et al., PRX Quantum 3, 010101 (2022)]. We further prove a two-parameter extension of the matrix inequality and show that two individually one-copy-undistillable NPT Werner states cannot activate each other's one-copy distillability. We also resolve the singular-value maximization problem associated with the two-ququart case.
Whether every entangled state with a non-positive partial transpose (NPT) is distillable reduces to the case of the Werner states ρ(d,α) = (I + αF)/(d² + αd), with F the swap, which for d ≥ 3 and −1/2 ≤ α < −1/d are NPT but one-copy undistillable. By the Schmidt-rank criterion of DiVincenzo et al. and Dür et al. (2000), as rewritten by P. Costa Rico (2025), ρ(d,α) is k-copy undistillable exactly when Σ_J α^|J| ‖Tr_J C‖² ≥ 0 for every vector of Schmidt rank at most two on k copies, written as a matrix C, where Tr_J traces out the copies in J. At α = −1/2 this inequality was proved independently for two copies by Fu et al., Fraser et al., Song and Chen, and Bharti et al. (2026), and for three copies by a computer-assisted proof in the companion record doi:10.5281/zenodo.23604343; if it held for every k, NPT bound entanglement would exist. We prove the inequality at α = −1/2 for every k and d ≥ 3 on the class of matrices C whose range is spanned by a vector f_0 of V_0 and a vector f_1 of V_1 and whose co-range (the range of C*) is arbitrary. Here V_c is spanned by |c⟩^⊗k and the k vectors obtained from it by replacing one factor with |2⟩, for orthonormal |0⟩, |1⟩, |2⟩. So no vector of this kind witnesses distillability of ρ(d,−1/2). We chose this class because, for generic f_0 and f_1, the projector onto the range has reduced states of rank three on every copy, while every known nonzero C with equality at α = −1/2 has a range with a reduced state of rank at most two on some copy. The proof reduces the claim by Schur complements to a quadratic form and then argues by induction on the number of copies. Python scripts in this record check the five rational identities of the proof symbolically; they do not involve k, and no other step uses a computer. We also prove that if the inequality holds for k − 1 copies but fails for k, then for some α > −1/2 it still holds, with equality at a nonzero C whose range has reduced states of rank at least three on every copy, as for generic members of the class. For two copies we prove that every nonzero C with equality at α = −1/2 has rank two and that the ranges of C and C* lie in products of two-dimensional single-copy subspaces. For general C the inequality remains open from four copies on.
Entanglement distillation is a fundamental task in quantum information theory. In this work, we prove that Werner states in arbitrary dimension are 222-copy distillable if and only if they are 111-copy distillable. This answers the longstanding open question of the 222-copy distillability of Werner states. This is an important step on determining whether every non-positive partial transpose (NPT) state is distillable, which remains one of the central open problems in the field of entanglement distillation.