Linear matrix maps for which positivity and complete positivity coincide

TL;DR

This paper identifies classes of linear matrix maps where positivity and complete positivity are equivalent, using structural conditions and a representation approach to simplify their analysis.

Contribution

It introduces a new class of linear matrix maps with coinciding positivity and complete positivity based on structural conditions and a representation framework.

Findings

Positivity and complete positivity coincide for certain structured linear matrix maps.

A representation of $*$-linear maps facilitates the analysis of positivity properties.

Sufficient conditions for positivity and complete positivity to coincide are established.

Abstract

By the Choi matrix criteria it is easy to determine if a specific linear matrix map is completely positive, but to establish whether a linear matrix map is positive is much less straightforward. In this paper we consider classes of linear matrix maps, determined by structural conditions on an associated matrix, for which positivity and complete positivity coincide. The basis of our proofs lies in a representation of $*$-linear matrix maps going back to work of R.D. Hill which enables us to formulate a sufficient condition in terms of surjectivity of certain bilinear maps.