Survey on Nonlocal Games and Operator Space Theory

TL;DR

This survey explores the deep connection between operator space theory and quantum nonlocality through nonlocal games, highlighting recent advances in Bell inequality violations, complexity, and entanglement quantification.

Contribution

It provides a comprehensive overview of how operator space norms relate to quantum nonlocality and summarizes recent results in this interdisciplinary area.

Findings

Large violations of Bell inequalities demonstrated

Complexity of quantum vs classical game values analyzed

Quantification of nonlocality for various entangled states

Abstract

This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states.