$C^{\ast }$-extreme points of entanglement breaking maps

Abstract

In this paper we study the $C^{\ast }$-convex set of unital entanglement breaking (EB-)maps on matrix algebras. General properties and an abstract characterization of $C^{\ast }$-extreme points are discussed. By establishing a Radon-Nikodym type theorem for a class of EB-maps we give a complete description of the $C^{\ast }$-extreme points. It is shown that a unital EB-map $\Phi :M_{d_1}\to M_{d_2}$ is $C^{\ast }$-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 $C^{\ast }$-convexity of the set of unital EB-maps.

References in corpus (2)

• Radon-Nikodym derivatives of quantum operations
• $C^{\ast }$-extreme points of positive operator valued measures and unital completely positive maps

Cited by in corpus (2)

• $C^{\ast }$-extreme contractive completely positive maps
• Degradable Strongly Entanglement Breaking Maps