Tsirelson's problem and Kirchberg's conjecture

TL;DR

This paper explores the connection between Tsirelson's problem and Kirchberg's QWEP conjecture, showing that the latter implies a positive answer to the former for all bipartite quantum scenarios, including spatial and spatiotemporal correlations.

Contribution

It establishes a link between the QWEP conjecture and Tsirelson's problem, extending the analysis to spatiotemporal correlations and formulating an extended version involving system steering.

Findings

Kirchberg's QWEP conjecture implies a positive answer to Tsirelson's problem.

Spatiotemporal correlations can reveal nonlocality even when spatial correlations are local.

An extended Tsirelson's problem is equivalent to the QWEP conjecture for each nontrivial Bell scenario.

Abstract

Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables located at different sites is assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products of C*-algebras would imply a positive answer to this question for all bipartite scenarios. This remains true also if one considers not only spatial correlations, but also spatiotemporal correlations, where each party is allowed to apply their measurements in temporal succession; we provide an example of a state together with observables such that ordinary spatial correlations are local, while the spatiotemporal correlations reveal nonlocality. Moreover, we find an extended version of Tsirelson's problem which, for each nontrivial Bell scenario, is equivalent to the QWEP conjecture. This extended version can be conveniently formulated in terms of steering the system of a third party. Finally, a comprehensive mathematical appendix offers background material on complete positivity, tensor products of C*-algebras, group C*-algebras, and some simple reformulations of the QWEP conjecture.