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Annales Henri Poincaré

, Volume 19, Issue 10, pp 2979–3005 | Cite as

Entanglement in Non-local Games and the Hyperlinear Profile of Groups

  • William Slofstra
  • Thomas Vidick
Article
  • 35 Downloads

Abstract

We relate the amount of entanglement required to play linear system non-local games near-optimally to the hyperlinear profile of finitely presented groups. By calculating the hyperlinear profile of a certain group, we give an example of a finite non-local game for which the amount of entanglement required to play \(\epsilon \)-optimally is at least \(\Omega (1/\epsilon ^k)\), for some \(k>0\). Since this function approaches infinity as \(\epsilon \) approaches zero, this provides a quantitative version of a theorem of the first author.

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Notes

Acknowledgements

As mentioned in Introduction, we are grateful to Narutaka Ozawa for suggesting the use of the Connes embedding trick and the beautiful line of argument now incorporated in Sect. 5; this led to a substantial improvement in our results. The first author also thanks Martino Lupini for helpful discussions. The second author is supported by NSF CAREER Grant CCF-1553477, AFOSR YIP Award Number FA9550-16-1-0495, a CIFAR Azrieli Global Scholar award, and the IQIM, an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  2. 2.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA

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