The tilted CHSH games: an operator algebraic classification

TL;DR

This paper develops an operator algebraic framework to classify tilted CHSH games, providing a simplified description of quantum advantage regions and a unique algebraic characterization of optimal states, advancing understanding of quantum correlations.

Contribution

It introduces a systematic operator algebraic method for analyzing tilted CHSH games, including a simplified anticommutation condition and a unique state classification.

Findings

Characterization of quantum advantage regions in tilted CHSH games

Simplified description of anticommutation requirements

Unique algebraic state maximizing quantum value

Abstract

We introduce a general systematic procedure for solving any binary-input binary-output game using operator algebraic techniques on the representation theory for the underlying group, which we then illustrate on the prominent class of tilted CHSH games: We derive for those an entire characterisation on the region exhibiting some quantum advantage and in particular derive a greatly simplified description for the required amount of anticommutation on observables (as being an essential ingredient in several adjacent articles). We further derive an abstract algebraic representation--free classification on the unique operator algebraic state maximising above quantum value. In particular the resulting operator algebraic state entails uniqueness for its corresponding correlation, including all higher and mixed moments. Finally the main purpose of this article is to provide above simplified description for the required amount of anticommutation and an abstract algebraic characterisation for their corresponding unique optimal state, both defining a key ingredient in upcoming work by the authors.