Bell and steering scenarios in terms of operator systems

TL;DR

This paper explores how operator systems can be used to describe non-locality, Bell, and steering scenarios, connecting quantum correlations with operator algebra structures and their approximations.

Contribution

It systematically links operator system tensor products with Tsirelson's correlation sets and NPA hierarchies, and applies these to Bell and steering inequalities.

Findings

Operator systems describe classes of quantum correlations.

Noncommuting cubes relate to steering assemblages.

Finite-dimensional approximations of noncommuting cubes are possible.

Abstract

The aim of this paper is to indicate possible applications of operator systems in qualitative description of varoius scenarios while studying non-locality. To this end we study in details the notion of generalized non-commuting cube. Following ideas of Fritz and Farenick-Kavruk-Paulsen-Todorov we show in systematic way that various classes of Tsirelson's correlation boxes as well as NPA hierarchies can be described by using various operator system tensor products of generalized non-commuting cubes. Moreover, we show also that noncommuting cubes can be applied for the description of steering assemblages. Next we study some aproximation properties of noncommuting cubes by finite dimensional models. Finaly, we indicate possibility to use the framework operator systems for studying Bell and steering inequalities.