Connes' embedding problem and Tsirelson's problem

TL;DR

This paper demonstrates the fundamental equivalence between Connes' embedding problem and Tsirelson's problem, linking quantum correlations with von Neumann algebra approximations, and showing that solutions to one imply solutions to the other.

Contribution

It establishes the equivalence between Connes' embedding problem and Tsirelson's problem, connecting quantum correlations with operator algebra conjectures.

Findings

Connes' embedding problem and Tsirelson's problem are essentially equivalent.

An affirmative answer to one problem implies a positive answer to the other.

The equivalence links quantum correlation sets with von Neumann algebra approximations.

Abstract

We show that Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg's QWEP conjecture) are essentially equivalent. Specifically, Tsirelson's problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C*-algebras. Connes' embedding problem asks whether any separable II$_1$ factor is a subfactor of the ultrapower of the hyperfinite II$_1$ factor. We show that an affirmative answer to Connes' question implies a positive answer to Tsirelson's. Conversely, a positve answer to a matrix valued version of Tsirelson's problem implies a positive one to Connes' problem.