On the Structure of Positive Maps between Matrix Algebras

We will be concerned with linear positive maps φ : Mm(C) → Mn(C). To fix notation we begin with setting up the notation and the relevant terminology (cf. [7]). We say that φ is positive if φ(A) is a positive element in Mn(C) for every positive matrix from Mm(C). If k ∈ N, then φ is said to be k-positive (respectively k-copositive) whenever [φ(Aij)] k i,j=1 (respectively [φ(Aji)] k i,j=1) is positive in Mk(Mn(C)) for every positive element [Aij ] k i,j=1 of Mk(Mm(C)). If φ is k-positive (respectively k-copositive) for every k ∈ N then we say that φ is completely positive (respectively completely copositive). Finally, we say that the map φ is decomposable if it has the form φ = φ1 + φ2 where φ1 is a completely positive map while φ2 is a completely copositive one. By P(m,n) we denote the set of all positive maps acting between Mm(C) and Mn(C) and by P1(m,n) – the subset of P(m,n) composed of all positive unital maps (i.e. such that φ(I) = I). Recall that P(m,n) has the structure of a convex cone while P1(m,n) is its convex subset. In the sequel we will use the notion of a face of a convex cone.