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

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.

32 pages

References in corpus (3)

• Operator system structures on ordered spaces
• Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory
• Quantum channels that preserve entanglement

Cited by in corpus (9)

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• The Ramsey property for Operator spaces and noncommutative Choquet simplices
• Completely Bounded Norms of $k$-positive Maps
• Operator Systems Generated by Projections
• Duality of Entanglement Norms
• Operator system characterizations of SIC-POVMs and mutually unbiased bases