Operator system structures on ordered spaces

TL;DR

This paper introduces minimal and maximal operator system structures on Archimedean order unit spaces, explores their properties, and characterizes entanglement breaking maps via these structures, advancing the understanding of operator systems.

Contribution

It constructs and analyzes minimal and maximal operator systems on ordered spaces, providing new characterizations and applications to entanglement breaking maps.

Findings

Operator systems OMIN(V) and OMAX(V) are constructed as analogues of minimal and maximal operator spaces.

Characterization of when an operator system is isomorphic to OMIN or OMAX.

Linear maps that are completely positive from OMIN(M_n) to OMAX(M_m) are exactly the entanglement breaking maps.

Abstract

Given an Archimedean order unit space (V,V^+,e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system S is completely boundedly isomorphic to either OMIN(S) or to OMAX(S). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(M_n) to OMAX(M_m) if and only if it is entanglement breaking.