Decomposability of extremal positive unital maps on M₂

Original · EN

A map ϕ:Mₘ()→ Mₙ() is decomposable if it is of the form ϕ=ϕ₁+ϕ₂ where ϕ₁ is a CP map while ϕ₂ 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:M₂()→ M₂() we construct concrete maps (not necessarily unital) ϕ₁ and ϕ₂ which give a decomposition of ϕ. We also show that in most cases this decomposition is unique.