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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Tsirelson’s problem and an embedding theorem for groups arising from non-local games
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by William Slofstra HTML | PDF
J. Amer. Math. Soc. 33 (2020), 1-56 Request permission

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.
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 4066471