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

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 OMINk(S) and a super k-maximal operator system OMAXk(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 OMINk(S) or to OMAXk(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 OMINk(Mn) to OMAXk(Mm) for some fixed k⩽min(n,m) if and only if it is a k-partially entanglement breaking map.