Bounding Quantum-Classical Separations for Classes of Nonlocal Games

Authors Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, Troy Lee

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Tom Bannink
  • CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Jop Briët
  • CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Harry Buhrman
  • CWI, University of Amsterdam & QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Farrokh Labib
  • CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Troy Lee
  • Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, Australia


We thank Peter Høyer, Serge Massar, and Henry Yuen for useful discussions. We thank Shravas Rao for providing a proof of one of the lemmas.

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Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee. Bounding Quantum-Classical Separations for Classes of Nonlocal Games. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum communication complexity
  • Nonlocal games
  • communication complexity
  • bounded separations
  • semidefinite programming
  • pseudorandomness
  • Gowers norms


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  1. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum Lower Bounds by Polynomials. Journal of the ACM, 48(4):778-797, 2001. Earlier version in FOCS'98. Google Scholar
  2. J.S. Bell. On the Einstein Podolsky Rosen Paradox. Physics, 1(3):195-200, 1964. Google Scholar
  3. Michael Ben-Or, Shafi Goldwasser, Joe Kilian, and Avi Wigderson. Multi-prover interactive proofs: how to remove intractability assumptions. In 20th Annual ACM Symposium on Theory of Computing (STOC '88), pages 113-131, 1988. Google Scholar
  4. Christian Borgs, Jennifer Chayes, László Lovász, Vera T Sós, and Katalin Vesztergombi. Counting graph homomorphisms. In Topics in discrete mathematics, pages 315-371. Springer, 2006. Google Scholar
  5. Michel Boyer. Extended GHZ n-player games with classical probability of winning tending to 0. arXiv, September 2004. URL:
  6. Mark Braverman, Konstantin Makarychev, Yury Makarychev, and Assaf Naor. The Grothendieck constant is strictly smaller than Krivine’s bound. In IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS '11), pages 453-462, 2011. Google Scholar
  7. Jop Briët, Harry Buhrman, Troy Lee, and Thomas Vidick. Multipartite entanglement in XOR games. Quantum Information &Computation, 13(3-4):334-360, 2013. Google Scholar
  8. Jop Briët and Thomas Vidick. Explicit lower and upper bounds on the entangled value of multiplayer XOR games. Communications in Mathematical Physics, 321(1):181-207, 2013. Google Scholar
  9. Moses Charikar, Konstantin Makarychev, and Yury Makarychev. Near-optimal Algorithms for Unique Games. In Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing, STOC '06, pages 205-214, New York, NY, USA, 2006. ACM. URL:
  10. Richard Cleve, Peter Høyer, Benjamin Toner, and John Watrous. Consequences and Limits of Nonlocal Strategies. In 19th Annual IEEE Conference on Computational Complexity (CCC 2004), 21-24 June 2004, Amherst, MA, USA, pages 236-249, 2004. URL:
  11. David Conlon, Hiêp Hàn, Yury Person, and Mathias Schacht. Weak quasi-randomness for uniform hypergraphs. Random Structures &Algorithms, 40(1):1-38, 2012. Google Scholar
  12. David Conlon and Joonkyung Lee. Finite reflection groups and graph norms. Advances in Mathematics, 315:130-165, 2017. Google Scholar
  13. Irit Dinur, Prahladh Harsha, Rakesh Venkat, and Henry Yuen. Multiplayer parallel repetition for expander games. arXiv preprint, 2016. URL:
  14. Irit Dinur, Elchanan Mossel, and Oded Regev. Conditional Hardness for Approximate Coloring. SIAM J. Comput., 39(3):843-873, 2009. URL:
  15. Alexander Grothendieck. Résumé de la théorie métrique des produits tensoriels topologiques (French). Bol. Soc. Mat. São Paulo, 8:1-79, 1953. Google Scholar
  16. Hamed Hatami. On generalizations of Gowers norms. PhD thesis, University of Toronto, 2009. Google Scholar
  17. Hamed Hatami. Graph norms and Sidorenko’s conjecture. Israel Journal of Mathematics, 175(1):125-150, 2010. Google Scholar
  18. Tsuyoshi Ito, Hirotada Kobayashi, Daniel Preda, Xiaoming Sun, and Andrew C. C. Yao. Generalized Tsirelson Inequalities, Commuting-Operator Provers, and Multi-prover Interactive Proof Systems. In Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity, CCC '08, pages 187-198. IEEE Computer Society, 2008. Google Scholar
  19. Julia Kempe, Oded Regev, and Ben Toner. Unique Games with Entangled Provers are Easy. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 457-466, 2008. URL:
  20. Iordanis Kerenidis, Sophie Laplante, Virginie Lerays, Jérémie Roland, and David Xiao. Lower bounds on information complexity via zero-communication protocols and applications. SIAM J. Comp., 44(5), 2015. Google Scholar
  21. Subhash Khot. On the Power of Unique 2-prover 1-round Games. In Proceedings of the Thiry-fourth Annual ACM Symposium on Theory of Computing (STOC '02), STOC '02, pages 767-775, New York, NY, USA, 2002. ACM. URL:
  22. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs? SIAM J. Comput., 37(1):319-357, 2007. URL:
  23. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci., 74(3):335-349, 2008. URL:
  24. Jean-Louis Krivine. Sur la constante de Grothendieck. C. R. Acad. Sci. Paris Sér. A-B, 284(8):A445–A446, 1977. Google Scholar
  25. Nati Linial and Adi Shraibman. Lower bounds in communication complexity based on factorization norms. Random Structures and Algorithms, 34:368-394, 2009. Google Scholar
  26. David Pérez-García, Michael Wolf, Carlos Palazuelos, Ignacio Villanueva, and Marius Junge. Unbounded violation of tripartite Bell inequalities. Communications in Mathematical Physics, 279:455, 2008. Google Scholar
  27. Ran Raz. Exponential separation of quantum and classical communication complexity. In 31st annual ACM symposium on theory of computing, pages 358-367, 1999. Google Scholar
  28. Yaoyun Shi and Yufan Zhu. Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement. SIAM J. Comput., 38(3):753-766, 2008. Google Scholar
  29. Terence Tao and Tamar Ziegler. The inverse conjecture for the Gowers norm over finite fields in low characteristic. Annals of Combinatorics, 16(1):121-188, 2012. Google Scholar
  30. Boris S. Tsirelson. Quantum analogues of the Bell Inequalities. The case of two spatially separated domains. J. Soviet Math., 36:557-570, 1987. Google Scholar
  31. Adam Bene Watts, Aram W Harrow, Gurtej Kanwar, and Anand Natarajan. Algorithms, Bounds, and Strategies for Entangled XOR Games. arXiv preprint, 2018. URL:
  32. M. Zukowski. Bell theorem involving all settings of measuring apparatus. Phys. Lett. A, 177(290), 1993. Google Scholar