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arXiv:quant-ph/0603015 · Physical Review A

General construction of noiseless networks detecting entanglement with the help of linear maps

Paweł Horodecki, Remigiusz Augusiak, Maciej Demianowicz

Published Nov 15, 2006• 21 citations
arXivHTMLPDF

Abstract

We present the general scheme for construction of noiseless networks detecting entanglement with the help of linear, hermiticity-preserving maps. We show how to apply the method to detect entanglement of an unknown state without its prior reconstruction. In particular, we prove there always exists noiseless network detecting entanglement with the help of positive, but not completely positive maps. Then the generalization of the method to the case of entanglement detection with arbitrary, not necessarily hermiticity-preserving, linear contractions on product states is presented.

DOI
10.1103/physreva.74.052323

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Bound entanglement detection in 4⊗44 \otimes 44⊗4 systems via generalized Choi maps
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Abstract The problem of entanglement detection is a long standing problem in quantum information theory. One of the primary procedures of detecting entanglement is to find the suitable positive but non-completely positive maps. Here we try to give a generic prescription to construct a positive map that can be useful for such scenarios. We study a class of positive maps arising from Lindblad structures. We show that two famous positive maps viz. transposition, reduction map and Choi map can be obtained as a special case of a class of positive maps having Lindblad structure. Generalizing the transposition map to a one parameter family we have used it to detect genuine multipartite entanglement. Finally being motivated by the negativity of entanglement, we have defined a similar measure for genuine multipartite entanglement.
In this paper, we discuss some general connections between the notions of positive map, weak majorization and entropic inequalities in the context of detection of entanglement among bipartite quantum systems. First, basing on the fact that any positive map can be written as the difference between two completely positive maps Λ=Λ 1 −Λ 2 , we propose a possible way to generalize the Nielsen–Kempe majorization criterion. Then, we present two methods of derivation of some general classes of entropic inequalities useful for the detection of entanglement. While the first one follows from the aforementioned generalized majorization relation and the concept of Schur-concave decreasing functions, the second is based on some functional inequalities. What is important is that, contrary to the Nielsen–Kempe majorization criterion and entropic inequalities, our criteria allow for the detection of entangled states with positive partial transposition when using indecomposable positive maps. We also point out that if a state with at least one maximally mixed subsystem is detected by some necessary criterion based on the positive map Λ, then there exist entropic inequalities derived from Λ (by both procedures) that also detect this state. In this sense, they are equivalent to the necessary criterion [ I ⊗Λ](ϱ AB )⩾0. Moreover, our inequalities provide a way of constructing multi-copy entanglement witnesses and therefore are promising from the experimental point of view. Finally, we discuss some of the derived inequalities in the context of the recently introduced protocol of state merging and the possibility of approximating the mean value of a linear entanglement witness.