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arXiv:1206.0504 · Journal of Physics A Mathematical and Theoretical

Optimal indecomposable witnesses without extremality or the spanning property

Kil-Chan Ha, Hoseog Yu

Published Sep 13, 2012• 9 citations
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Abstract

One of the interesting problems of optimal indecomposable entanglement witnesses is whether there exists an optimal indecomposable entanglement witness which neither has the spanning property nor is associated with an extremal positive linear map. Here, we answer this question in the negative by examining the extremality of the positive linear maps constructed by Qi and Hou (2011 J. Phys. A: Math. Theor. 44 215305).

DOI
10.1088/1751-8113/45/39/395307

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One of the interesting problems of optimal indecomposable entanglement witnesses is whether there exists an optimal indecomposable entanglement witness which neither has the spanning property nor is associated with an extremal positive linear map. Here, we answer this question in the negative by examining the extremality of the positive linear maps constructed by Qi and Hou (2011 J. Phys. A: Math. Theor. 44 215305).
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Optimal PPTES witnesses for states in Cn⊗Cn\mathbb C^n\otimes \mathbb C^nCn⊗Cn
One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces ${\mathcal{H}}_{m,n}={\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ such that the product vectors obeying $\ensuremath{\langle}e,f|W|e,f\ensuremath{\rangle}=0$ do not span Hm,n{\mathcal{H}}_{m,n}Hm,n​. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of $2\ensuremath{\bigotimes}n$ optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces Hm,n{\mathcal{H}}_{m,n}Hm,n​ with $m,n\ensuremath{\ge}3$.
We examine various notions related to the optimality for entanglement witnesses arising from Choi-type positive linear maps. We found examples of optimal entanglement witnesses which are nondecomposable but are not ``nondecomposable optimal entanglement witnesses'' in the sense of Lewenstein, Kraus, Cirac, and Horodecki [Phys. Rev. A 62, 052310 (2000)]. We suggest using the terms PPTES witness and optimal PPTES witness in place of ``nondecomposable entanglement witness'' and ``nondecomposable optimal entanglement witnesses'' in order to avoid possible confusion. Here, PPTES refers entangled states with positive partial transposes. We also found examples of nonextremal optimal entanglement witnesses which are indecomposable.
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 consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entangled state can be detected by one of these witnesses, and this witness detects a unique set of entangled states among those. Therefore, they provide a minimal set of witnesses to detect any kind of entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entangled states with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one-parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps in three-dimensional matrix algebra.