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Connes' embedding problem and Tsirelson's problem

### 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 factor is a subfactor of the ultrapower of the hyperfinite II 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.

© 2011 American Institute of Physics

Received 08 September 2010
Accepted 18 October 2010
Published online 06 January 2011

Acknowledgments:
This work was supported in part by Spanish grants I-MATH, MTM2008-01366, S2009/ESP-1594, the European projects QUEVADIS and CORNER, DFG grant We1240/12-1 and National Science Foundation grant DMS-0901457. VBS would like to thank Fabian Furrer for stimulating discussions.

Article outline:

I. INTRODUCTION
II. TSIRELSON'S PROBLEM
III. CONNES' EMBEDDING PROBLEM
IV. NON-SIGNALLING OPERATOR SYSTEM AND TENSOR NORMS
V. CONNES' EMBEDDING PROBLEM EQUALS TSIRELSON's PROBLEM
VI. SUMMARY