Exposed positive maps: a sufficient condition

TL;DR

This paper establishes a sufficient condition for positive maps in matrix algebras to be exposed, enhancing understanding of their extremal properties and optimality, with specific analysis on decomposable maps.

Contribution

It introduces a new sufficient condition for exposed positive maps and explores its necessity in decomposable maps, advancing the theoretical framework of positive map extremality.

Findings

Exposed positive maps form a dense subset of extremal maps.

The paper provides a sufficient condition for a positive map to be exposed.

For decomposable maps, the condition is also necessary.

Abstract

Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a sufficient condition for a positive map to be exposed. This is an analog of a spanning property which guaranties that a positive map is optimal. We analyze a class of decomposable maps for which this condition is also necessary.