Synchronous correlation matrices and Connes' embedding conjecture

TL;DR

This paper explores the relationship between synchronous quantum correlation matrices and Connes' embedding conjecture, establishing their equivalence and implications for graph parameters and non-local games.

Contribution

It proves the equivalence of Connes' embedding conjecture with the equality of two families of synchronous quantum correlation matrices and explores consequences for graph theory.

Findings

Connes' embedding conjecture is equivalent to the equality of two families of synchronous quantum correlation matrices.

If true, the tracial rank and projective rank are equal for every graph.

Results extend to more general non-local games.

Abstract

In a recent paper, the concept of synchronous quantum correlation matrices was introduced and these were shown to correspond to traces on certain C*-algebras. In particular, synchronous correlation matrices arose in their study of various versions of quantum chromatic numbers of graphs and other quantum versions of graph theoretic parameters. In this paper we develop these ideas further, focusing on the relations between synchronous correlation matrices and microstates. We prove that Connes' embedding conjecture is equivalent to the equality of two families of synchronous quantum correlation matrices. We prove that if Connes' embedding conjecture has a positive answer, then the tracial rank and projective rank are equal for every graph. We then apply these results to more general non-local games.