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


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.

This is a preview of subscription content, log in to check access.


  1. 1.

    Cavaleri, M.: Algorithms and quantifications in amenable and sofic groups. Ph.D. thesis, Università di Roma La Sapienza (2016)

  2. 2.

    Cleve, R., Liu, L., Slofstra, W.: Perfect commuting-operator strategies for linear system games. J. Math. Phys. 58, 012202 (2017)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. 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. 4.

    Coudron, M., Natarajan, A.: The parallel-repeated magic square game is rigid. Tech. report (2016). arXiv:1609.06306

  5. 5.

    Coladangelo, A.W.: Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH. Quantum Inf. Comput. 17(9), 831–865 (2017)

    MathSciNet  Google Scholar 

  6. 6.

    Connes, A.: Classification of injective factors cases II1, II, III, 1. Ann. Math. 104(1), 73–115 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Cornulier, Y.: Sofic profile and computability of Cremona groups. Mich. Math. J. 62(4), 823–841 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  8. 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)

  9. 9.

    Fritz, T.: On infinite-dimensional state spaces. J. Math. Phys. 54(5), 052107 (2013)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Hadwin, D., Shulman, T.: Stability of group relations under small Hilbert–Schmidt perturbations. J. Funct. Anal. 275(4), 761–792 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Junge, M., Oikhberg, T., Palazuelos, C.: Reducing the number of inputs in nonlocal games. J. Math. Phys. 57(10), 102203 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Junge, M., Palazuelos, C.: Large violation of bell inequalities with low entanglement. Commun. Math. Phys. 306(3), 695–746 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Mančinska, L.: Maximally Entangled State in Pseudo-Telepathy Games. Computing with New Resources, pp. 200–207. Springer, Berlin (2014)

    Google Scholar 

  15. 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)

  16. 16.

    Ostrev, D., Vidick, T.: Entanglement of approximate quantum strategies in XOR games. Quantum Inf. Comput. 18(7 & 8), 0617–0631 (2018)

    MathSciNet  Google Scholar 

  17. 17.

    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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Paulsen, V.I., Severini, S., Stahlke, D., Todorov, I.G., Winter, A.: Estimating quantum chromatic numbers. J. Funct. Anal. 270(6), 2188–2222 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Slofstra, W.: Lower bounds on the entanglement needed to play XOR non-local games. J. Math. Phys. 52(10), 102202 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Slofstra, W.: The set of quantum correlations is not closed (2017). arXiv:1703.08618

Download references


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).

Author information



Corresponding author

Correspondence to Thomas Vidick.

Additional information

Communicated by David Perez-Garcia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation