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

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 $\mathrm{\Phi }:M_{d_1}\to M_{d_2}$ 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.