The Jamiołkowski Isomorphism and a Simplified Proof for the Correspondence Between Vectors Having Schmidt Number k and k-Positive Maps

Abstract

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.