Tsirelson's problem and an embedding theorem for groups arising from non-local games

TL;DR

This paper demonstrates that the commuting-operator and tensor-product models for quantum correlations are not equivalent by providing counterexamples using non-local games, and shows that certain related problems are undecidable.

Contribution

It proves the existence of non-local games with perfect commuting-operator strategies but no perfect tensor-product strategies, and embeds arbitrary finitely-presented groups into solution groups.

Findings

Counterexamples to Tsirelson's problem using non-local games.

Embedding of any finitely-presented group into solution groups.

Undecidability of the perfect strategy problem for linear system games.

Abstract

Tsirelson's problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.