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Journal of Physics Conference Series

Spectral conditions for positive maps and entanglement witnesses

Dariusz Chruściński, Andrzej Kossakowski

Published Mar 1, 2011• 6 citations
Publisher

Abstract

We provide partial classification of entanglement witnesses and positive maps which is based on a family of spectral conditions. Interestingly many well known examples of positive maps (e.g. transposition, reduction map, the family of Choi maps) belong to the class constructed via spectral conditions.
DOI
10.1088/1742-6596/284/1/012017

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We construct a class of quantum bipartite $d\ensuremath{\bigotimes}d$ states which are positive under partial transposition (PPT states). This class is invariant under the maximal commutative subgroup of U(d)\mathrm{U}(d)U(d) and contains as special cases many well-known examples of PPT states. States from our class provide criteria for testing the indecomposability of positive maps. Such maps are crucial for constructing entanglement witnesses.
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.
We present a multi-parameter family of positive maps between spaces of arbitrary but finite dimensions. This framework facilitates the construction of Entanglement Witnesses (EWs) specifically designed for systems living in [Formula: see text] dimensions. A sufficient condition for positivity is presented. Interestingly, it is shown that all EWs constructed this way are equivalent to a single realignment-like criterion which for [Formula: see text] and [Formula: see text] are different is in general stronger than the original realignment criterion. We illustrate effectiveness of this criterion considering examples of Positive Partial Transpose entangled states in [Formula: see text] and [Formula: see text] dimensions.