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Modern Physics Letters B

A special case on the distillability problem of two-copy n×n Werner states

Huilin Chen

Published Jan 29, 2024• 1 citation
Publisher

Abstract

In this paper, we delve into a nuanced aspect of the distillability problem within the realm of entanglement theory, specifically examining the distillability of anti-diagonal [Formula: see text] entangled Werner states under the framework of local operations and classical communication (LOCC). To address this, we scrutinize a conjecture central to the distillability issue, initially focusing on its implications in 4-dimensional and 5-dimensional contexts. Through rigorous calculations, we establish that these cases align with the conjecture. Subsequently, we broaden our investigation to encompass [Formula: see text] matrices, demonstrating that both even and odd n-dimensional anti-diagonal matrices are in concordance with the conjecture, thus contributing to a deeper understanding of the distillability problem in quantum entanglement.

DOI
10.1142/s0217984924502099

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In this paper, we delve into a nuanced aspect of the distillability problem within the realm of entanglement theory, specifically examining the distillability of anti-diagonal [Formula: see text] entangled Werner states under the framework of local operations and classical communication (LOCC). To address this, we scrutinize a conjecture central to the distillability issue, initially focusing on its implications in 4-dimensional and 5-dimensional contexts. Through rigorous calculations, we establish that these cases align with the conjecture. Subsequently, we broaden our investigation to encompass [Formula: see text] matrices, demonstrating that both even and odd n-dimensional anti-diagonal matrices are in concordance with the conjecture, thus contributing to a deeper understanding of the distillability problem in quantum entanglement.
The distillability conjecture of two-copy 4 by 4 Werner states is one of the main open problems in quantum information. We prove two special cases of the conjecture. The first case occurs when two 4 by 4 matrices A, B are both unitarily equivalent to block diagonal matrices with 2 by 2 blocks. The second case occurs when B is unitarily equivalent to either -A or the transpose of -A. Plus, we propose a simplified version of the distillability conjecture when both A and B are matrices with distinct eigenvalues.
The distillability conjecture for two-copy four-by-four Werner states has been an open problem in quantum information for years. We investigate the conditions under which the conjectured inequality becomes an equality. For all known cases where the conjecture has been verified, we characterize the saturation conditions and show that equality forces the matrices AAA and BBB to be two-by-two block-diagonal. In particular, several previously obtained partial results, including the cases of one normal matrix, unitary similarity between BBB and −A-A−A or −AT-A^T−AT, and anti-diagonal block structures, are reduced to this common block-diagonal structure. We also employ a manifold optimization method, which provides numerical evidence that the two-by-two block-diagonal structure is essential for saturating the inequality. Furthermore, we prove that the identified saturation points are critical points of the objective function on the constraint manifold.
The pure entangled state is of vital importance in the field of quantum information. The process of asymptotically extracting pure entangled states from many copies of mixed states via local operations and classical communication is called entanglement distillation. The entanglement distillability problem, which is a long-standing open problem, asks whether such process exists. The 2-copy undistillability of 4×44\times44×4 undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix A⊗I+I⊗BA\otimes I + I \otimes BA⊗I+I⊗B does not exceed (3d−4)/d2(3d-4)/d^2(3d−4)/d2 with A,BA,BA,B traceless d×dd\times dd×d matrices and ∣∣A∣∣F2+∣∣B∣∣F2=1/d||A||_F^2+||B||_F^2=1/d∣∣A∣∣F2​+∣∣B∣∣F2​=1/d when d=4d=4d=4. The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices AAA and BBB are normal. In this paper, we prove that the conjecture holds when one of matrices AAA and BBB is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.
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.
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.
Some progress is reported on conditions for convertibility among bipartite 2x2 entangled states: An inconvertibility condition related to the rank of an entangled state is given that it is impossible to convert to an entangled state with lower rank under separable operations; a particular set of local operations and classical communication (LOCC) is used to analyze convertibility of three subclasses of states - Werner states, Bell diagonal states and maximally entangled mixed states (MEMS). It is conjectured that MEMS may lie on the bottom of entangled state ordering for given entanglement of formation. A plausible way is suggested of systematically calculating convertibility in a general subclass of bipartite states whose density matrices are defined to be diagonal in a common basis. The set of LOCC adopted in this work is argued to be generalizable to provide sufficient conditions for convertibility among a large range of general 2x2 entangled states.