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arXiv:1008.5056 · Journal of Physics A Mathematical and Theoretical

On structural physical approximations and entanglement breaking maps

Remigiusz Augusiak, J Bae, Ł Czekaj, Maciej Lewenstein

Published Apr 11, 2011• 33 citations

Abstract

Very recently, a conjecture saying that the so-called structural physical approximations (SPAs) to optimal positive maps (optimal entanglement witnesses) give entanglement breaking (EB) maps (separable states) has been posed (Korbicz et al 2008 Phys. Rev. A 78 062105). The main purpose of this contribution is to explore this subject. First, we extend the set of entanglement witnesses supporting the conjecture. Then, we ask whether SPAs constructed from other than the depolarizing channel maps also lead to EB maps and show that in general this is not the case. On the other hand, we prove an interesting fact that for any positive map Λ, there exists an EB channel Φ such that the SPA of Λ constructed with the aid of Φ is again an EB channel. Finally, we ask similar questions in the case of continuous variable systems. We provide a simple way of constructing SPA and prove that in the case of the transposition map it gives the EB channel.

DOI: 10.1088/1751-8113/44/18/185308

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Very recently, a conjecture saying that the so-called structural physical approximations (SPAs) to optimal positive maps (optimal entanglement witnesses) give entanglement breaking (EB) maps (separable states) has been posed (Korbicz et al 2008 Phys. Rev. A 78 062105). The main purpose of this contribution is to explore this subject. First, we extend the set of entanglement witnesses supporting the conjecture. Then, we ask whether SPAs constructed from other than the depolarizing channel maps also lead to EB maps and show that in general this is not the case. On the other hand, we prove an interesting fact that for any positive map Λ, there exists an EB channel Φ such that the SPA of Λ constructed with the aid of Φ is again an EB channel. Finally, we ask similar questions in the case of continuous variable systems. We provide a simple way of constructing SPA and prove that in the case of the transposition map it gives the EB channel.

The structural physical approximation (SPA) to a positive map is considered to be one of the most important methods to detect entanglement in the real physical world. We first show that an arbitrary entanglement witness (EW)

W

can be constructed from a separable density matrix $\ensuremath{\sigma}$ in the form of $W=\ensuremath{\sigma}\ensuremath{-}{c}_{\ensuremath{\sigma}}I$, where ${c}_{\ensuremath{\sigma}}$ is a non-negative number and

I

is the identity matrix. Following the general form of EWs from separable states, we show a sufficient condition and a sufficient and necessary condition in low dimensions of that SPAs to positive maps do not define entanglement-breaking channels. We show that either the SPA of an EW or the SPA of the partial transposition of the EW in low dimensions is an entanglement-breaking channel. We give sufficient conditions of violating the SPA conjecture [Phys. Rev. A 78, 062105 (2008)]. Our results indicate that the SPA conjecture is independent of whether or not positive maps are optimal.

It was conjectured that the structural physical approximation (SPA) of an optimal entanglement witness is separable (or equivalently, that the SPA of an optimal positive map is entanglement breaking). This conjecture was disproved, first for indecomposable maps and more recently for decomposable maps. The arguments in both cases are sketched along with important related results. This review includes background material on topics including entanglement witnesses, optimality, duality of cones, decomposability, and the statement and motivation for the SPA conjecture so that it should be accessible for a broad audience.

In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain indecomposable optimal witnesses that have no spanning property. These also provide examples that support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).

Based on the general form of entanglement witnesses constructed from separable states, we first show a sufficient condition of violating the structural physical approximation (SPA) conjecture [Phys. Rev. A 78, 062105 (2008)]. Then we discuss the SPA conjecture for decomposable entanglement witnesses. Moreover, we make geometric illustrations of the connection between entanglement witnesses and the sets of quantum states, separable states, and entangled states comparing with planes and vectors in Euclidean space.

We provide a class of indecomposable entanglement witnesses. In $4\ifmmode\times\else\texttimes\fi{}4$ case, it reproduces the well-known Breuer-Hall witness. We prove that these witnesses are optimal and atomic, i.e., they are able to detect the ``weakest'' quantum entanglement encoded into states with positive partial transposition. Equivalently, we provide a construction of indecomposable atomic maps in the algebra of $2k\ifmmode\times\else\texttimes\fi{}2k$ complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et al. [Phys. Rev. A 78, 062105 (2008)].

In 2008, the conjecture that structural physical approximations to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, in [K.-C. Ha and S.-H. Kye, Separable states with unique decompositions, arXiv:1210.1088v3], Ha and Kye proposed a decomposable entanglement witness whose SPA is entangled and argued that it is optimal. In this note, which is based on a comment to the latter work [R. Augusiak et al., Comment on "Separable states with unique decompositions", arXiv:1304.2040v1], we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking optimality of entanglement witnesses developed already in [M. Lewenstein et al., Phys. Rev. A 62, 052310 (2000)], however, hardly exploited so far in the literature.