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arXiv:1205.0592

Optimal PPTES witnesses for states in $\mathbb C^n\otimes \mathbb C^n$

Kil-Chan Ha

Published May 3, 2012• 1 citation
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Abstract

Recently, X. Qi and J. Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated to indecomposable positive linear maps, but it is not checked whether partial transposes of these witnesses are also optimal. We show that partial transposes of these entanglement witnesses have spanning property, and these witnesses are indeed optimal PPTES witnesses (non-decomposable optimal entanglement witnesses).

DOI
10.48550/arxiv.1205.0592

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Optimal PPTES witnesses for states in Cn⊗Cn\mathbb C^n\otimes \mathbb C^nCn⊗Cn
Recently, X. Qi and J. Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated to indecomposable positive linear maps, but it is not checked whether partial transposes of these witnesses are also optimal. We show that partial transposes of these entanglement witnesses have spanning property, and these witnesses are indeed optimal PPTES witnesses (non-decomposable optimal entanglement witnesses).
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).
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.
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 construct optimal PPTES witnesses to detect 3⊗33\otimes 33⊗3 PPT entangled edge states of type (6,8)(6,8)(6,8) constructed recently \cite{kye_osaka}. To do this, we consider positive linear maps which are variants of the Choi type map involving complex numbers, and examine several notions related to optimality for those entanglement witnesses. Through the discussion, we suggest a method to check the optimality of entanglement witnesses without the spanning property.
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).