Mathematical Programming

, Volume 162, Issue 1–2, pp 431–463 | Cite as

Linear conic formulations for two-party correlations and values of nonlocal games

  • Jamie Sikora
  • Antonios VarvitsiotisEmail author
Full Length Paper


In this work we study the sets of two-party correlations generated from a Bell scenario involving two spatially separated systems with respect to various physical models. We show that the sets of classical, quantum, no-signaling and unrestricted correlations can be expressed as projections of affine sections of appropriate convex cones. As a by-product, we identify a spectrahedral outer approximation to the set of quantum correlations which is contained in the first level of the Navascués, Pironio and Acín (NPA) hierarchy and also a sufficient condition for the set of quantum correlations to be closed. Furthermore, by our conic formulations, the value of a nonlocal game over the sets of classical, quantum, no-signaling and unrestricted correlations can be cast as a linear conic program. This allows us to show that a semidefinite programming upper bound to the classical value of a nonlocal game introduced by Feige and Lovász is in fact an upper bound to the quantum value of the game and moreover, it is at least as strong as optimizing over the first level of the NPA hierarchy. Lastly, we show that deciding the existence of a perfect quantum (resp. classical) strategy is equivalent to deciding the feasibility of a linear conic program over the cone of completely positive semidefinite matrices (resp. completely positive matrices). By specializing the results to synchronous nonlocal games, we recover the conic formulations for various quantum and classical graph parameters that were recently derived in the literature.


Quantum correlations Nonlocal games Completely positive semidefinite cone Completely positive cone Linear conic programming Quantum graph parameters Semidefinite programming relaxations 

Mathematics Subject Classification

90C22 90C90 90C25 81P40 81P45 



The authors would like to thank the referees for carefully reading the paper and for their useful comments. Furthermore, the authors thank S. Burgdorf, M. Laurent, L. Mančinska, T. Piovesan, D. E. Roberson, S. Upadhyay, T. Vidick and Z. Wei for useful discussions. A.V. is supported in part by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13. J.S. is supported in part by NSERC Canada. Research at the Centre for Quantum Technologies at the National University of Singapore is partially funded by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant “Random numbers from quantum processes,” (MOE2012-T3-1-009).


  1. 1.
    Aspect, A., Grangier, P., Roger, G.: Experimental realization of Einstein–Podolsky–Rosen–Bohm–Gedanken experiment: a new violation of Bell’s inequalities. Phys. Rev. Lett. 49, 91–94 (1982)CrossRefGoogle Scholar
  2. 2.
    Barvinok, A.: A Course in Convexity. American Mathematical Society, Providence (2002)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bell, J.S.: On the Einstein–Podolsky–Rosen paradox. Physics 1(3), 195–200 (1964)Google Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MOS-SIAM Series on Optimization (2001)Google Scholar
  5. 5.
    Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, Singapore (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Berta, M., Fawzi, O., Scholz, V.B.: Quantum bilinear optimization. arXiv:1506.08810 (2015)
  7. 7.
    Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86(2), 419 (2014)CrossRefGoogle Scholar
  8. 8.
    Burgdorf, S., Laurent, M., Piovesan, T.: On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings. arXiv:1502.02842 (2015)
  9. 9.
    Cameron, P.J., Montanaro, A., Newman, M.W., Severini, S., Winter, A.: On the quantum chromatic number of a graph. Electr. J. Comb. 14(1). arXiv:quant-ph/0608016 (2007)
  10. 10.
    Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23(15), 880–884 (1969)CrossRefGoogle Scholar
  11. 11.
    Cleve, R., Høyer, P., Toner, B., Watrous, J.: Consequences and limits of nonlocal strategies. In: Proceedings of the 19th Annual IEEE Conference on Computational Complexity, pp. 236–249 (2004)Google Scholar
  12. 12.
    Cleve, R., Slofstra, W., Unger, F., Upadhyay, S.: Perfect parallel repetition theorem for quantum XOR proof systems. Comput. Complex. 17(2), 282–299 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Colbeck, R.: Quantum and relativistic protocols for secure multi-party computation. Ph.D. thesis, Trinity College, University of Cambridge (2006)Google Scholar
  14. 14.
    de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dinur, I., Steurer, D.: Analytical approach to parallel repetition. In: Proceedings of the 46th ACM Symposium on Theory of Computing (2014)Google Scholar
  16. 16.
    Dinur, I., Steurer, D., Vidick, T.: A parallel repetition theorem for entangled projection games. In: Proceedings of the 29th IEEE Conference on Computational Complexity, pp. 197–208 (2014)Google Scholar
  17. 17.
    Dykema, K.J., Paulsen, V.: Synchronous correlation matrices and Connes’ embedding conjecture. J. Math. Phys. 57, 015214 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67(6), 661–663 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Feige, U., Lovász, L.: Two-prover one-round proof systems: their power and their problems. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pp. 733–744. ACM (1992)Google Scholar
  20. 20.
    Freedman, S.J., Clauser, J.F.: Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938–941 (1972)CrossRefGoogle Scholar
  21. 21.
    Fritz, T.: Polyhedral duality in Bell scenarios with two binary observables. J. Math. Phys. 53, 072202 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fritz, T.: Tsirelson’s problem and Kirchberg’s conjecture. Rev. Math. Phys. 24(5), 1250012 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ji, Z.: Binary constraint system games and locally commutative reductions. arXiv:1310.3794 (2013)
  24. 24.
    Kempe, J., Regev, O., Toner, B.: Unique games with entangled provers are easy. SIAM J. Comput. 39(7), 3207–3229 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lasserre, J.B.: New approximations for the cone of copositive matrices and its dual. Math. Program. 144(1–2), 265–276 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Laurent, M., Piovesan, T.: Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone. SIAM J. Optim. 25(4), 2461–2493 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mančinska, L., Roberson, D.E.: Note on the correspondence between quantum correlations and the completely positive semidefinite cone. Unpublished manuscript, available at (2014)
  28. 28.
    Mančinska, L., Roberson, D.E.: Quantum homomorphisms. J. Comb. Theory Ser. B 118, 228–267 (2016)Google Scholar
  29. 29.
    Mančinska, L., Roberson, D.E., Varvitsiotis, A.: On deciding the existence of perfect entangled strategies for nonlocal games. Chicago J. Theor. Comput. Sci. arXiv:1506.07429 (2016)
  30. 30.
    Maxfield, J.E., Minc, H.: On the matrix equation \(X^{\prime }X = A\). Proc. Edinb. Math. Soc. (Ser. 2) 13(02), 125–129 (1962)CrossRefzbMATHGoogle Scholar
  31. 31.
    Navascués, M., Pironio, S., Acín, A.: A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys. 10(7), 073013 (2008)CrossRefGoogle Scholar
  32. 32.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  33. 33.
    Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, California Institute of Technology (2000)Google Scholar
  34. 34.
    Paulsen, V.I., Severini, S., Stahlke, D., Todorov, I.G., Winter, A.: Estimating quantum chromatic numbers. J. Funct. Anal. 270(6), 2188–2222 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Roberson, D.E.: Conic formulations of graph homomorphisms. J. Algebraic Comb. 43(4), 877–913 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sikora, J., Varvitsiotis, A., Wei, Z.: On the minimum dimension of a Hilbert space needed to generate a quantum correlation. arXiv:1507.00213 (2015)
  37. 37.
    Tsirelson, B.S.: Quantum analogues of the Bell inequalities: the case of two spatially separated domains. J. Sov. Math. 36, 557–570 (1987)CrossRefGoogle Scholar
  38. 38.
    Upadhyay, S.: Quantum Information and Variants of Interactive Proof Systems. Ph.D. thesis, University of Waterloo (2011)Google Scholar
  39. 39.
    Watrous, J.: Theory of quantum information, lecture notes. (2011)

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore
  2. 2.MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654SingaporeSingapore
  3. 3.School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

Personalised recommendations