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arXiv:1204.6596 · Physical Review A

Optimality for indecomposable entanglement witnesses

Kil-Chan Ha, Seung-Hyeok Kye

Published Sep 4, 2012• 30 citations
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Abstract

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.

DOI
10.1103/physreva.86.034301

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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.
Optimal PPTES witnesses for states in Cn⊗Cn\mathbb C^n\otimes \mathbb C^nCn⊗Cn
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).
An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES's). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES's, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM's). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW's and PM's. Finally, we illustrate our results by constructing optimal EW acting on $H={C}^{2}\ensuremath{\bigotimes}{C}^{4}.$ The corresponding PM's constitute examples of PM's with minimal ``qubit'' domains, or---equivalently---minimal Hermitian conjugate codomains.
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.