The Super Operator System Structures and their applications in Quantum Entanglement Theory

An operator system $\text{cl}S$ with unit $e$, can be viewed as an Archimedean order unit space $(\text{cl}S,\text{cl}{S}^{+},e)$. Using this Archimedean order unit space, for a fixed $k\in \text{bb}N$ we construct a super k-minimal operator system OMIN${}_k(\text{cl}S)$ and a super k-maximal operator system OMAX${}_k(\text{cl}S)$, which are the general versions of the minimal operator system OMIN$(\text{cl}S)$ and the maximal operator system OMAX$(\text{cl}S)$ 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 $\text{cl}S$ is completely boundedly isomorphic to either OMIN${}_k(\text{cl}S)$ or to OMAX${}_k(\text{cl}S)$. 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 $k\le \min (n,m)$ if and only if it is a k-partially entanglement breaking map.