The Communication Complexity of Non-signaling Distributions

  • Julien Degorre
  • Marc Kaplan
  • Sophie Laplante
  • Jérémie Roland
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5734)


We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a,b distributed according to some pre-specified joint distribution p(a,b|x,y). Our results apply to any non-signaling distribution, that is, those where Alice’s marginal distribution does not depend on Bob’s input, and vice versa.

By introducing a simple new technique based on affine combinations of lower-complexity distributions, we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. The lower bounds we obtain can be expressed as linear programs (or SDPs for quantum communication). We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions.

We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds.

Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution.


Boolean Function Quantum Correlation Communication Complexity Quantum Communication Bell Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Yao, A.C.C.: Some complexity questions related to distributive computing. In: Proc. 11th STOC, pp. 209–213 (1979)Google Scholar
  2. 2.
    Kushilevitz, E., Nisan, N.: Communication complexity. Cambridge University Press, New York (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    Maudlin, T.: Bell’s inequality, information transmission, and prism models. In: Biennal Meeting of the Philosophy of Science Association, pp. 404–417 (1992)Google Scholar
  4. 4.
    Brassard, G., Cleve, R., Tapp, A.: Cost of Exactly Simulating Quantum Entanglement with Classical Communication. Phys. Rev. Lett. 83, 1874–1877 (1999); quant-ph/9901035CrossRefGoogle Scholar
  5. 5.
    Steiner, M.: Towards quantifying non-local information transfer: finite-bit non-locality. Phys. Lett. A 270, 239–244 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Toner, B.F., Bacon, D.: Communication Cost of Simulating Bell Correlations. Phys. Rev. Lett. 91, 187904 (2003)CrossRefGoogle Scholar
  7. 7.
    Cerf, N.J., Gisin, N., Massar, S., Popescu, S.: Simulating Maximal Quantum Entanglement without Communication. Phys. Rev. Lett. 94(22), 220403 (2005)CrossRefGoogle Scholar
  8. 8.
    Degorre, J., Laplante, S., Roland, J.: Classical simulation of traceless binary observables on any bipartite quantum state. Phys. Rev. A 75(012309) (2007)Google Scholar
  9. 9.
    Regev, O., Toner, B.: Simulating quantum correlations with finite communication. In: Proc. 48th FOCS, pp. 384–394 (2007)Google Scholar
  10. 10.
    Shi, Y., Zhu, Y.: Tensor norms and the classical communication complexity of nonlocal quantum measurement. SIAM J. Comput. 38(3), 753–766 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jones, N.S., Masanes, L.: Interconversion of nonlocal correlations. Phys. Rev. A 72, 052312 (2005)CrossRefGoogle Scholar
  12. 12.
    Barrett, J., Pironio, S.: Popescu-Rohrlich correlations as a unit of nonlocality. Phys. Rev. Lett. 95, 140401 (2005)CrossRefGoogle Scholar
  13. 13.
    Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. Random Struct. Algorithms 34(3), 368–394 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Navascues, M., Pironio, S., Acin, A.: A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys. 10(7), 073013 (2008)CrossRefGoogle Scholar
  15. 15.
    Randall, C.H., Foulis, D.J.: Operational statistics and tensor products. In: Interpretations and Foundations of Quantum Theory, Volume Interpretations and Foundations of Quantum Theory, pp. 21–28. Wissenschaftsverlag, BibliographischesInstitut (1981)Google Scholar
  16. 16.
    Foulis, D.J., Randall, C.H.: Empirical logic and tensor products. In: Interpretations and Foundations of Quantum Theory, Volume Interpretations and Foundations of Quantum Theory, pp. 1–20. Wissenschaftsverlag, BibliographischesInstitut (1981)Google Scholar
  17. 17.
    Kläy, M., Randall, C.H., Foulis, D.J.: Tensor products and probability weights. Int. J. Theor. Phys. 26(3), 199–219 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wilce, A.: Tensor products in generalized measure theory. Int. J. Theor. Phys. 31(11), 1915–1928 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75(3), 032304 (2007)Google Scholar
  20. 20.
    Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. SIAM J. Comput. 35(4), 787–803 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tsirelson, B.S.: Zapiski Math. Inst. Steklov (LOMI) 142, 174–194 (1985); English translation in Quantum analogues of the Bell inequalities. The case of two spatially separated domains. J. Soviet Math. 36, 557–570 (1987)Google Scholar
  22. 22.
    Linial, N., Mendelson, S., Schechtman, G., Shraibman, A.: Complexity measures of sign matrices. Combinatorica 27, 439–463 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964)Google Scholar
  24. 24.
    Tsirelson, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4(2), 93–100 (1980)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yao, A.C.C.: On the power of quantum fingerprinting. In: Proc. 35th STOC, pp. 77–81 (2003)Google Scholar
  26. 26.
    Gavinsky, D., Kempe, J., de Wolf, R.: Strengths and weaknesses of quantum fingerprinting. In: Proc. 21st CCC, pp. 288–295 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Julien Degorre
    • 1
  • Marc Kaplan
    • 2
  • Sophie Laplante
    • 2
  • Jérémie Roland
    • 3
  1. 1.Laboratoire d’Informatique de GrenobleCNRSFrance
  2. 2.LRIUniversité Paris-SudFrance
  3. 3.NEC Laboratories AmericaUSA

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