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THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  14 January 2019

WILLIAM SLOFSTRA
WILLIAM SLOFSTRA*
Affiliation:
Department of Pure Mathematics and Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada; [email protected]

Abstract

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We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.

MSC classification

Primary: 81P45: Quantum information, communication, networks
Secondary: 20F05: Generators, relations, and presentations
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 7 , 2019 , e1
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits noncommercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author 2019

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