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Linear Algebra and its Applications

A new criterion and a special class of k-positive maps

Jinchuan Hou, Chi-Kwong Li, Yiu‐Tung Poon, Xiaofei Qi, Nung-Sing Sze

Published Mar 12, 2014• 20 citations
Publisher
DOI
10.1016/j.laa.2014.02.008

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Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and, more generally, as witnesses for states having Schmidt number not greater than k. Such maps and witnesses are related to k-positive maps, and their properties may be investigated by making use of the Jamiołkowski isomorphism. In this article we review the properties of this isomorphism, noting that there are actually two related mappings bearing that name. As a new result, we give a simplified proof for the correspondence between vectors having Schmidt number k and k-positive maps and thus for the Jamiołkowski criterion for complete positivity. Another consequence is a special case of a result by Choi, namely that k-positivity implies complete positivity, if k is the dimension of the smaller one of the Hilbert spaces on which the operators act.
We present a new class of entangled bipartite states. These remain positive under partial transposition, and hence live outside the jurisdiction of the Peres‐Horodecki separability criterion. Their inseparability is demonstrated using Choi‐type indecomposable positive maps. It is shown that these states are not isolated, but occupy a finite volume in the bipartite state space. It is further shown that in the space of positive maps there exists a neighborhood around the Choi map with the property that every map in this neighborhood is indecomposable.