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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Cavaleri, M.: Algorithms and quantifications in amenable and sofic groups. Ph.D. thesis, Università di Roma La Sapienza (2016)
Cleve, R., Liu, L., Slofstra, W.: Perfect commuting-operator strategies for linear system games. J. Math. Phys. 58, 012202 (2017)
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
Coudron, M., Natarajan, A.: The parallel-repeated magic square game is rigid. Tech. report (2016). arXiv:1609.06306
Coladangelo, A.W.: Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH. Quantum Inf. Comput. 17(9), 831–865 (2017)
Connes, A.: Classification of injective factors cases II1, II, III, 1. Ann. Math. 104(1), 73–115 (1976)
Cornulier, Y.: Sofic profile and computability of Cremona groups. Mich. Math. J. 62(4), 823–841 (2013)
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)
Fritz, T.: On infinite-dimensional state spaces. J. Math. Phys. 54(5), 052107 (2013)
Hadwin, D., Shulman, T.: Stability of group relations under small Hilbert–Schmidt perturbations. J. Funct. Anal. 275(4), 761–792 (2018)
Junge, M., Oikhberg, T., Palazuelos, C.: Reducing the number of inputs in nonlocal games. J. Math. Phys. 57(10), 102203 (2016)
Junge, M., Palazuelos, C.: Large violation of bell inequalities with low entanglement. Commun. Math. Phys. 306(3), 695–746 (2011)
Junge, M., Palazuelos, C., Pérez-García, D., Villanueva, I., Wolf, M.M.: Unbounded violations of bipartite bell inequalities via operator space theory. Commun. Math. Phys. 300(3), 715–739 (2010)
Mančinska, L.: Maximally Entangled State in Pseudo-Telepathy Games. Computing with New Resources, pp. 200–207. Springer, Berlin (2014)
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)
Ostrev, D., Vidick, T.: Entanglement of approximate quantum strategies in XOR games. Quantum Inf. Comput. 18(7 & 8), 0617–0631 (2018)
Pérez-García, D., Wolf, M.M., Palazuelos, C., Villanueva, I., Junge, M.: Unbounded violation of tripartite bell inequalities. Commun. Math. Phys. 279(2), 455–486 (2008)
Paulsen, V.I., Severini, S., Stahlke, D., Todorov, I.G., Winter, A.: Estimating quantum chromatic numbers. J. Funct. Anal. 270(6), 2188–2222 (2016)
Slofstra, W.: Lower bounds on the entanglement needed to play XOR non-local games. J. Math. Phys. 52(10), 102202 (2011)
Slofstra, W.: The set of quantum correlations is not closed (2017). arXiv:1703.08618
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).
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