Rank properties of exposed positive maps

TL;DR

This paper investigates the structure of positive linear maps between finite-dimensional operator spaces, identifying specific forms that are exposed points of the cone of all positive maps and characterizing their rank properties.

Contribution

It proves that maps of the form $P$ are exposed points and characterizes the rank behavior of all exposed positive maps.

Findings

Maps of the form $P$ are exposed points of the cone.

Exposed positive maps are either rank 1 non-increasing or have rank greater than 1 on all one-dimensional projections.

Provides a structural understanding of the extremal properties of positive maps.

Abstract

Let $\cK$ and $\cH$ be finite dimensional Hilbert spaces and let $\fP$ denote the cone of all positive linear maps acting from $\fB(\cK)$ into $\fB(\cH)$. We show that each map of the form $\phi(X)=AXA^*$ or $\phi(X)=AX^TA^*$ is an exposed point of $\fP$. We also show that if a map $\phi$ is an exposed point of $\fP$ then either $\phi$ is rank 1 non-increasing or $\rank\phi(P)>1$ for any one-dimensional projection $P\in\fB(\cK)$.