Decomposability of extremal positive unital maps on $M_2$

Abstract

A map $ϕ:M_m(\bC)\to M_n(\bC)$ is decomposable if it is of the form $\varphi ={\varphi }_1+{\varphi }_2$ where ${\varphi }_1$ is a CP map while ${\varphi }_2$ is a co-CP map. It is known that if $m=n=2$ then every positive map is decomposable. Given an extremal unital positive map $ϕ:M_2(\bC)\to M_2(\bC)$ we construct concrete maps (not necessarily unital) ${\varphi }_1$ and ${\varphi }_2$ which give a decomposition of $\varphi$. We also show that in most cases this decomposition is unique.

9 pages