Tsirelson's problem and an embedding theorem for groups arising from non-local games
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 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.
Keywords
Cite
@article{arxiv.1606.03140,
title = {Tsirelson's problem and an embedding theorem for groups arising from non-local games},
author = {William Slofstra},
journal= {arXiv preprint arXiv:1606.03140},
year = {2020}
}
Comments
Update to match published version, including a new background section on quantum correlation sets, fixes to the definition of pictures in Section 6 (the previous definition using smooth curves allowed some pathological behaviour), and other minor changes throughout. 66 pages, 36 figures