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Decomposability of extremal positive unital maps on $M_2$

Wladyslaw A. Majewski·Marcin Marciniak
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Abstract

A map $ϕ:M_m(\bC)\to M_n(\bC)$ is decomposable if it is of the form $ϕ=ϕ_1+ϕ_2$ where $ϕ_1$ is a CP map while $ϕ_2$ 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) $ϕ_1$ and $ϕ_2$ which give a decomposition of $ϕ$. We also show that in most cases this decomposition is unique.

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Wladyslaw A. Majewski, Marcin Marciniak. 2005-10-21. Decomposability of extremal positive unital maps on $M_2$. https://arxiv.org/abs/math/0510005

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