Connes' embedding problem and Tsirelson's problem

By Marius Junge, Miguel Navascués, Carlos Palazuelos, David Pérez-García, Volkher B. Scholz, Reinhard F. Werner. 2010

199 citations recorded by the source corpus.

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 II1 factor is a subfactor of the ultrapower of the hyperfinite II1 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.

Subfields: Algebra and Number Theory, Atomic and Molecular Physics, and Optics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd1be212dc37cb8ae695f32700f36d8e9ec691eb

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Citations in the corpus

QUANTUM CHROMATIC NUMBERS VIA OPERATOR SYSTEMS (2013, background): “But we now know, thanks to the work [12], that equality of all matricial outcomes for the two different versions is equivalent to Connes’ Embedding Problem.”
Grothendieck's Theorem, past and present (2011, background): “The Connes-Kirchberg problem As it turns out (see [68, 37]) Tsirelson’s problem is (essentially) equivalent to one of the most famous ones in von Neumann algebra theory going back to Connes’s paper [26].”
Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture (2013, background): “Other very recent discoveries include the fact that Connes’ embedding conjecture is related to Tsirelson’s problem, a major open problem in Quantum Information Theory (see [49], [66], and [93]) and that it is connected to the “minimal” and “commuting” tensor products [73] of some group operator syst”
A Combinatorial Approach to Nonlocality and Contextuality (2012, background): “This question is a generalization of Tsirelson’s problem [43,57], and can be asked more precisely like this: 5.”
Nuclearity Related Properties in Operator Systems (2011, background): “In [20] it was shown that when the actors are Alice and Bob (that is, in the bipartite scenario) the question is reduced to whether”
Tensor Products of C*-Algebras and Operator Spaces (2020, methodology): “The proof of this equivalence was completed in several steps, in the papers [95] and [138], the final step being taken by Ozawa in [191].”
Large Violation of Bell Inequalities with Low Entanglement (2010, background): “ the result follows easily just reasoning by duality on real elements (Ma;b x;y ) x;y;a;b. Remark 6.8. The space NSG(N;K) represents the formalization of the comments after Theorem 3.2 . Actually, in [34] it is shown the utility of this space to study the set of probability distributions when we ass”