Skip to main content
SpringerLink
Log in

Classical versus quantum communication in XOR games

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games when Alice and Bob are allowed to use a limited amount (c bits) of one-way classical communication compared to their value when they are allowed to use the same amount of one-way quantum communication (c qubits). The key quantity here is the ratio between the quantum and classical value. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the previous quotient is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (PNAS 113(12):3191–3196, 2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information c. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical versus quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. XOR games are often introduced replacing \(\{-1,1\}\) by \(\{0,1\}\) and replacing the product by the XOR of the variables. In that case, our value of the game translates into the bias, the additional probability over \(\frac{1}{2}\) of winning the game.

References

  1. Bell, J.S.: On the Einstein–Poldolsky–Rosen paradox. Physics 1, 195 (1964)

    Article  Google Scholar 

  2. Briet, J., Vidick, T.: Explicit lower and upper bounds on the entangled value of multiplayer XOR games. Commun. Math. Phys. 321(1), 181–207 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Brukner, C., Zukowski, M., Pan, J.-W., Zeilinger, A.: Violation of Bell’s inequality: criterion for quantum communication complexity advantage. Phys. Rev. Lett. 92, 127901 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  4. Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86(2), 419 (2014)

    Article  ADS  Google Scholar 

  5. Buhrman, H., Cleve, R., Massar, S., de Wolf, R.: Non-locality and communication complexity. Rev. Mod. Phys. 82, 665 (2010)

    Article  ADS  Google Scholar 

  6. Buhrman, H., Czekaj, L., Grudka, A., Horodecki, M., Horodecki, P., Markiewicz, M., Speelman, F., Strelchuk, S.: Quantum communication complexity advantage implies violation of a Bell inequality. PNAS 113(12), 3191–3196 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Buhrman, H., Regev, O., Scarpa, G., de Wolf, R.: Near-Optimal and Explicit Bell Inequality Violations. In: IEEE Conference on Computational Complexity, pp. 157–166 (2011)

  8. Cleve, R., Hoyer, P., Toner, B., Watrous, J.: Consequences and limits of nonlocal strategies. In: Proceedings of 19th IEEE Conference on Computational Complexity (CCC’04), pp. 236–249. IEEE Computer Society (2004)

  9. Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland, Amsterdam (1993)

    MATH  Google Scholar 

  10. Ekert, A., Renner, R.: The ultimate physical limits of privacy. Nature 507(7493), 443–447 (2014)

    Article  ADS  Google Scholar 

  11. Gavinsky, D., Kempe, J., Kerenidis, I., Raz, R., de Wolf, R.: Exponential separations for one-way quantum communication complexity, with applications to cryptography. In: Proceedings of the 39th ACM STOC, pp. 516–525 (2007)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Junge, M., Palazuelos, C.: CB-norm estimates for maps between noncommutative \(L_p\)-spaces and quantum channel theory. Int. Math. Res. Not. 3, 875–925 (2016)

    Article  MATH  Google Scholar 

  15. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  16. Laplante, S., Lauriere, M., Nolin, A., Roland, J., Senno, G.: Robust Bell inequalities from communication complexity. In: 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016), volume 61 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 5:1–5:24 (2016)

  17. Latala, R.: Some estimates of norms of random matrices. Proc. Am. Math. Soc. 133(5), 1273–1282 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Linial, N., Mendelson, S., Schechtman, G., Shraibman, A.: Complexity measures of sign matrices. Combinatorica 27(4), 439–463 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. Random Struct. Algorithms 34, 368–394 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Palazuelos, C., Vidick, T.: Survey on nonlocal games and operator space theory. J. Math. Phys. 57, 015220 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Palazuelos, C., Yin, Z.: Large bipartite Bell violations with dichotomic measurements. Phys. Rev. A 92, 052313 (2015)

    Article  ADS  Google Scholar 

  22. Pérez-García, D., Wolf, M.M., Palazuelos, C., Villanueva, I., Junge, M.: Unbounded violation of tripartite Bell inequalities. Comm. Math. Phys. 279(2), 455–486 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Raz, R.: Exponential separation of quantum and classical communication complexity. In: Proceedings of 31st ACM STOC, pp. 358–367 (1999)

  24. Regev, O., Klartag, B.: Quantum one-way communication can be exponentially stronger than classical communication. In: Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing (STOC’11), pp. 31–40 (2011)

  25. Schechtman, G.: More on embeddings subspaces of \(L_p\) in \(\ell _r^n\). Compos. Math. 61, 159–170 (1987)

    Google Scholar 

  26. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tsirelson, B.S.: Some results and problems on quantum Bell-type inequalities. Hadron. J. Supp. 8(4), 329–345 (1993)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Funding was provided by National Science Foundation (Grant No. NSF DMS-1201886) and Ministerio de Economía y Competitividad (Grant Nos. RYC-2012-10449, MTM2014-54240-P, SEV-2015-0554).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, IL, 61891, USA

    Marius Junge

  2. Instituto de Ciencias Matemáticas (ICMAT), Campus de Cantoblanco, 28049, Madrid, Spain

    Carlos Palazuelos

  3. Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Carlos Palazuelos & Ignacio Villanueva

  4. Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040, Madrid, Spain

    Ignacio Villanueva

Authors
  1. Marius Junge
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos Palazuelos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ignacio Villanueva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ignacio Villanueva.

Additional information

Marius Junge is partially supported by NSF DMS-1201886. Carlos Palazuelos is partially supported by the Spanish “Ramón y Cajal program” (RYC-2012-10449). Marius Junge and Carlos Palazuelos are partially supported by the Spanish “Severo Ochoa Programe” for Centres of Excellence (SEV-2015-0554). Carlos Palazuelos and Ignacio Villanueva are partially supported by the Grants MTM2014-54240-P, funded by Spanish MINECO, and QUITEMAD+-CM, S2013/ICE-2801, funded by Comunidad de Madrid.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Junge, M., Palazuelos, C. & Villanueva, I. Classical versus quantum communication in XOR games. Quantum Inf Process 17, 117 (2018). https://doi.org/10.1007/s11128-018-1883-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-018-1883-0

Keywords

Search

Navigation