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Tsirelson’s problem and an embedding theorem for groups arising from non-local games


Author: William Slofstra
Journal: J. Amer. Math. Soc. 33 (2020), 1-56
MSC (2010): Primary 20F06, 20F10, 81P40; Secondary 81P13, 81R15
DOI: https://doi.org/10.1090/jams/929
Published electronically: September 27, 2019
MathSciNet review: 4066471
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Abstract:

Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to the Connes embedding problem, remains open.

The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.


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Additional Information

William Slofstra
Affiliation: Institute for Quantum Computing and Department of Pure Mathematics, University of Waterloo, Canada
MR Author ID: 841088
Email: weslofst@uwaterloo.ca

Received by editor(s): July 1, 2016
Received by editor(s) in revised form: July 23, 2018
Published electronically: September 27, 2019
Additional Notes: This research was partially supported by NSERC grant number 2018-03968.
Article copyright: © Copyright 2019 American Mathematical Society