Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory

TL;DR

This paper explores the structure of minimal and maximal operator spaces and systems in quantum information, linking them to entanglement detection and providing new criteria for assessing quantum state properties.

Contribution

It establishes connections between operator space norms and entanglement measures, introducing new tools for detecting k-positive maps and Schmidt number.

Findings

Matrix norms for k-minimal spaces relate to entanglement detection tools.

Cones of positive elements in operator systems correspond to k-block positive operators.

Norm-based criteria for Schmidt number testing generalize existing separability criteria.

Abstract

We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.