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arXiv:1109.2300 · Physical Review A

Characterization of optimal entanglement witnesses

Xiaofei Qi, Jinchuan Hou

Published Feb 27, 2012• 27 citations
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Abstract

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).

DOI
10.1103/physreva.85.022334

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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).
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)].
If one wants to establish optimality of a given bipartite entanglement witness, the current standard approach is to check whether it has the spanning property. Although this is not necessary for optimality, it is most often satisfied in practice, and for small enough dimensions or sufficiently structured witnesses this criterion can be checked by hand. In this work we introduce a novel characterization of the spanning property via entanglement-breaking channels, which in turn leads to a new sufficient criterion for optimality. This criterion amounts to just checking the kernel of some bipartite state. It is slightly weaker than the spanning property, but it is a lot easier to test for -- by hand as well as numerically -- and it applies to almost all witnesses which are known to have the spanning property. A second criterion is derived from this, where one can simply compute the expectation value of the given witness on a maximally entangled state. Finally, this approach implies new spectral constraints on witnesses as well as on positive maps.
Recently, Qi and Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated with indecomposable positive linear maps, but whether partial transposes of these witnesses are also optimal has not been evaluated. We show that partial transposes of these entanglement witnesses have the spanning property and that these witnesses are indeed optimal witnesses detecting positive-partial- transposed entangled states (nondecomposable optimal entanglement witnesses).
Optimal PPTES witnesses for states in Cn⊗Cn\mathbb C^n\otimes \mathbb C^nCn⊗Cn
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.