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

TL;DR

This paper introduces super operator system structures, OMIN$_k$ and OMAX$_k$, generalizing minimal and maximal systems, and applies them to characterize k-partially entanglement breaking maps in quantum information theory.

Contribution

It develops new super operator system structures and explores their properties, linking them to quantum entanglement breaking maps, advancing the understanding of operator systems in quantum information.

Findings

Defined super k-minimal and super k-maximal operator systems.

Characterized when an operator system is isomorphic to these super systems.

Connected these structures to k-partially entanglement breaking maps.

Abstract

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