A completely positive map associated with a positive map

TL;DR

The paper demonstrates that every positive map between finite-dimensional operator algebras can be expressed as a scalar multiple of a trace minus a completely positive map, providing new insights into the structure of positive maps.

Contribution

It establishes a canonical form for positive maps as scalar multiples of trace minus completely positive maps, and derives conditions for C-positivity in various mapping cones.

Findings

Positive maps are scalar multiples of trace minus completely positive maps.

Necessary and sufficient conditions for C-positivity are provided.

Results apply to k-positive maps and large classes of mapping cones.

Abstract

We show that each positive map from B(K) to B(H) with K and H finite dimensional Hilbert spaces is a scalar multiple of a map of the form $Tr - \psi$ with $\psi$ completely positive. This is used to give necessary and sufficient conditions for maps to be C-positive for a large class of mapping cones; in particular we apply the results to k-positive maps.