C⁎-extreme entanglement breaking maps on operator systems

Abstract

Let E denote the set of all unital entanglement breaking (UEB) linear maps defined on an operator system S⊂Md and, mapping into Mn. As it turns out, the set E is not only convex in the classical sense but also in a quantum sense, namely it is C⁎-convex. The main objective of this article is to describe the C⁎-extreme points of this set E. By observing that every EB map defined on the operator system S dilates to a positive map with commutative range and also extends to an EB map on Md, we show that the C⁎-extreme points of the set E are precisely the UEB maps that are maximal in the sense of Arveson ([1] and [2]) and that they are also exactly the linear extreme points of the set E with commutative range. We also determine their explicit structure, thereby obtaining operator system generalizations of the analogous structure theorem and the Krein-Milman type theorem given in [8]. As a consequence, we show that C⁎-extreme (UEB) maps in E extend to C⁎-extreme UEB maps on the full algebra. Finally, we obtain an improved version of the main result in [8], which contains various characterizations of C⁎-extreme UEB maps between the algebras Md and Mn.