Indecomposable Positive Maps in Matrix Algebras

Abstract

We prove that Choi's map in M3 cannot be written as the sum of a 2-positive map and a 2-copositive map. We also provide other examples of positive maps in Mn which cannot be written as the sum of an n-positive map and a 2-copositive map.

Cones of positive maps and their duality relations

“…That is, we call them "2-supercopositive maps". On the other hand, the families D 2,2 3 and D 2,2 4 , which we would call 2-decomposable, appeared many times in the context of atomic maps [15,50,51]. An element of L (H) is called atomic iff it does not belong to D 2,2 (H).…”

Construction of 3 otimes 3 entangled edge states with positive partial transposes

“…Every decomposable map is positive, that is, sends positive semi-definite matrices into themselves, but the converse is not true. There are many examples of indecomposable positive linear maps in the literature [4], [6], [10], [11], [12], [19], [20], [26], [28], [30], [31], [32], [33]. We have shown in [23] that every face of the cone D is of the form…”

A Class of Extremal Positive Maps in $3 \times 3$ Matrix Algebras

“…A map cp e P(M n ) Is said to be decomposable if <p can be a sum of a completely positive map and a completely copositive map. As a new candidate for the previous basic problem, Tanahasi and Tomiyama [12] have introduced the following concept;…”

“…That this is not the case for higher dimensional algebras was shown by Choi [3] at first by an example of indecomposable maps in M 3 (C). Recently, Kye [6], Tanahasi and Tomiyama [12], and the author [8,9] have studied strong positive indecomposable maps in M n (C) such that they can not be decomposed into a sum of a 2-positive map and a 2-copositive map. Another approach to the set P(M n ) is to study extremal points in P(M n ).…”