Connes' embedding problem and Tsirelson's problem

Journal Article · Sat Jan 15 04:00:00 EST 2011 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3514538· OSTI ID:21501239

Junge, M; Palazuelos, C ^[1]; Navascues, M; Perez-Garcia, D ^[2]; Scholz, V B; Werner, R F ^[3]

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1. Department of Mathematics, University of Illinois at Urbana-Champaign, Illinois 61801-2975 (United States)
2. Departamento de Analisis Matematico and IMI, Universidad Complutense de Madrid, 28040, Madrid (Spain)
3. Institut fuer Theoretische Physik, Leibniz Universitaet Hannover, Appelstr. 2, 30167 Hannover (Germany)

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{sup *}-algebras. Connes' embedding problem asks whether any separable II{sub 1} factor is a subfactor of the ultrapower of the hyperfinite II{sub 1} factor. We show that an affirmative answer to Connes' question implies a positive answer to Tsirelson's. Conversely, a positive answer to a matrix valued version of Tsirelson's problem implies a positive one to Connes' problem.

• OSTI ID: 21501239
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 1 Vol. 52; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

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