The super operator system structures and their applications in quantum entanglement theory

Abstract

An operator system S with unit e, can be viewed as an Archimedean order unit space ( S , S + , e ) . Using this Archimedean order unit space, for a fixed k ∈ N we construct a super k-minimal operator system OMIN k ( S ) and a super k-maximal operator system OMAX k ( S ) , which are the general versions of the minimal operator system OMIN ( S ) and the maximal operator system OMAX ( S ) introduced recently, such that for k = 1 we obtain the equality, respectively. We develop some of the key properties of these super operator systems and make some progress on characterizing when an operator system S is completely boundedly isomorphic to either OMIN k ( S ) or to OMAX k ( S ) . Then we apply these concepts to the study of k-partially entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN k ( M n ) to OMAX k ( M m ) for some fixed k ⩽ min ( n , m ) if and only if it is a k-partially entanglement breaking map.