Exposed faces for decomposable positive linear maps arising from completely positive maps

TL;DR

This paper characterizes when certain faces of the cone of decomposable positive linear maps are exposed, based on the structure of rank one matrices in a subspace of 2 by n matrices.

Contribution

It provides a precise criterion involving rank one matrices for the exposedness of faces in the cone of decomposable positive maps.

Findings

A face of the cone of completely positive maps is exposed if and only if rank one matrices form a subspace with zero.

The orthogonal complement of the face is spanned by rank one matrices.

The characterization links geometric properties of matrix subspaces to positivity map structures.

Abstract

Let $D$ be a space of $2\times n$ matrices. Then the face of the cone of all completely positive maps from $M_2$ into $M_n$ given by $D$ is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in $D$ forms a subspace of $D$ together with zero and $D^\perp$ is spanned by rank one matrices.