Entanglement in Non-local Games and the Hyperlinear Profile of Groups
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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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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).
- 1.Cavaleri, M.: Algorithms and quantifications in amenable and sofic groups. Ph.D. thesis, Università di Roma La Sapienza (2016)Google Scholar
- 3.Cleve, R., Mittal, R.: Characterization of binary constraint system games. In: Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 8572, pp. 320–331. Springer, Berlin (2014). arXiv:1209.2729
- 4.Coudron, M., Natarajan, A.: The parallel-repeated magic square game is rigid. Tech. report (2016). arXiv:1609.06306
- 8.Chao, R., Reichardt, B.W., Sutherland, C., Vidick, T.: Test for a large amount of entanglement, using few measurements. In: Proceedings of the 2017 Conference on Innovations in Theoretical Computer Science (ITCS) (2017)Google Scholar
- 15.Natarajan, A., Vidick, T.: A quantum linearity test for robustly verifying entanglement. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (New York, NY, USA), STOC, pp. 1003–1015. ACM (2017)Google Scholar
- 20.Slofstra, W.: The set of quantum correlations is not closed (2017). arXiv:1703.08618