Almost synchronous quantum correlations

TL;DR

This paper demonstrates that correlations nearly synchronous in an $ ext{l}_1$ sense can be approximated by strategies involving maximally entangled states, simplifying the analysis of nonlocal games and their properties.

Contribution

It extends previous results on exactly synchronous correlations to almost synchronous cases, providing dimension-independent approximation guarantees.

Findings

Almost synchronous correlations can be approximated by convex combinations of maximally entangled strategies.

The approximation quality is independent of Hilbert space dimension.

This facilitates analysis of nonlocal games and rigidity properties.

Abstract

The study of quantum correlation sets initiated by Tsirelson in the 1980s and originally motivated by questions in the foundations of quantum mechanics has more recently been tied to questions in quantum cryptography, complexity theory, operator space theory, group theory, and more. Synchronous correlation sets introduced in [Paulsen et. al, JFA 2016] are a subclass of correlations that has proven particularly useful to study and arises naturally in applications. We show that any correlation that is almost synchronous, in a natural $\ell_1$ sense, arises from a state and measurement operators that are well-approximated by a convex combination of projective measurements on a maximally entangled state. This extends a result of [Paulsen et. al, JFA 2016] which applies to exactly synchronous correlations. Crucially, the quality of approximation is independent of the dimension of the Hilbert spaces or of the size of the correlation. Our result allows one to reduce the analysis of many classes of nonlocal games, including rigidity properties, to the case of strategies using maximally entangled states which are generally easier to manipulate.