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

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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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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Correspondence to Thomas Vidick.

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Communicated by David Perez-Garcia.

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Slofstra, W., Vidick, T. Entanglement in Non-local Games and the Hyperlinear Profile of Groups. Ann. Henri Poincaré 19, 2979–3005 (2018). https://doi.org/10.1007/s00023-018-0718-y

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