$C^*$-extreme points of entanglement breaking maps

TL;DR

This paper characterizes the $C^*$-extreme points of unital entanglement breaking maps on matrix algebras, providing a complete description and a noncommutative Krein-Milman theorem, advancing the understanding of quantum channels.

Contribution

It offers a full characterization of $C^*$-extreme points of unital EB-maps using Radon-Nikodym theorems and Choi-rank conditions, and establishes a noncommutative Krein-Milman theorem.

Findings

$C^*$-extreme unital EB-maps have Choi-rank equal to the output dimension.

Complete description of $C^*$-extreme points via Radon-Nikodym type theorem.

Derived a noncommutative Krein-Milman theorem for EB-maps.

Abstract

In this paper we study the $C^*$-convex set of unital entanglement breaking (EB-)maps on matrix algebras. General properties and an abstract characterization of $C^*$-extreme points are discussed. By establishing a Radon-Nikodym type theorem for a class of EB-maps we give a complete description of the $C^*$-extreme points. It is shown that a unital EB-map $\Phi:M_{d_1}\to M_{d_2}$ is $C^*$-extreme if and only if it has Choi-rank equal to $d_2$. Finally, as a direct consequence of the Holevo form of EB-maps, we derive a noncommutative analogue of the Krein-Milman theorem for $C^*$-convexity of the set of unital EB-maps.