Advertisement

What Can Be Computed without Communications?

  • Heger Arfaoui
  • Pierre Fraigniaud
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7355)

Abstract

This paper addresses the following 2-player problem. Alice (resp., Bob) receives a boolean x (resp., y) as input, and must return a boolean a (resp., b) as output. A game between Alice and Bob is defined by a pair (δ,f) of boolean functions. The objective of Alice and Bob playing game (δ,f) is, for every inputs x and y, to output values a and b, respectively, satisfying δ(a,b) = f(x,y), in absence of any communication between the two players.It is known that, for xor-games, that is, games equivalent, up to individual reversible transformations, to a game (δ,f) with δ(a,b) = a ⊕ b, the ability for the players to use entangled quantum bits (qbits) helps: there exist a distributed protocol for the chsh game, using quantum correlations, for which the probability that the two players produce a successful output is higher than the maximum probability of success of any classical distributed protocol for that game, even when using shared randomness.

In this paper, we show that, apart from xor-games, quantum correlations does not help, in the sense that, for every such game, there exists a classical protocol (using shared randomness) whose probability of success is at least as large as the one of any protocol using quantum correlations. This result holds for both worst case and average case analysis. It is achieved by considering a model stronger than quantum correlations, the non-signaling model, for which we show that, if the game is not an xor-game, then shared randomness is a sufficient resource for the design of optimal protocols. These results provide an invitation to revisit the theory of distributed checking, a.k.a. distributed verification. Indeed, the literature dealing with this theory is mostly focusing on decision functions δ equivalent to the and-operator. This paper demonstrates that such a decision function may not well be suited for taking benefit of the computational power of quantum correlations.

Keywords

Boolean Function Success Probability Quantum Correlation Probabilistic Guarantee Quantum Protocol 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barrett, J., Linden, N., Massar, S., Pironio, S., Popescu, S., Roberts, D.: Nonlocal correlations as an information-theoretic resource. Physical Review A 71(2), 1–11 (2005)CrossRefGoogle Scholar
  2. 2.
    Barrett, J., Pironio, S.: Popescu-Rohrlich correlations as a unit of nonlocality. Phys. Rev. Lett. 95(14) (2005)Google Scholar
  3. 3.
    Bell, J.S.: On the Einstein-Podolsky-Rosen paradox. Physics 1(3), 195–200 (1964)Google Scholar
  4. 4.
    Buhrman, H., Cleve, R., Massar, S., de Wolf, R.: Non-locality and communication complexity. Reviews of Modern Physics 82, 665–698 (2010)CrossRefGoogle Scholar
  5. 5.
    Cirel’son, B.S.: Quantum generalizations of bell’s inequality. Letters in Math. Phys. 4(2), 93–100 (1980)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Physical Review Letters 23(15), 880–884 (1969)CrossRefGoogle Scholar
  7. 7.
    Das Sarma, A., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed verification and hardness of distributed approximation. In: 43rd ACM Symp. on Theory of Computing, STOC (2011)Google Scholar
  8. 8.
    Dupuis, F., Gisin, N., Hasidim, A., Allan Méthot, A., Pilpel, H.: No nonlocal box is universal. J. Math. Phys. 48(082107) (2007)Google Scholar
  9. 9.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Physical Review 47(10), 777–780 (1935)zbMATHCrossRefGoogle Scholar
  10. 10.
    Fraigniaud, P., Korman, A., Peleg, D.: Local distributed decision. In: 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 708–717 (2011)Google Scholar
  11. 11.
    Fraigniaud, P., Rajsbaum, S., Travers, C.: Locality and Checkability in Wait-Free Computing. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 333–347. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Fraigniaud, P., Rajsbaum, S., Travers, C.: Universal distributed checkers and orientation-detection tasks (submitted, 2012)Google Scholar
  13. 13.
    Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distributed Computing 22, 215–233 (2010)CrossRefGoogle Scholar
  14. 14.
    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM J. Comput. 24(6), 1259–1277 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Foundations of Physics 24(3), 379–385 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Heger Arfaoui
    • 1
  • Pierre Fraigniaud
    • 1
  1. 1.LIAFACNRS and University Paris DiderotFrance

Personalised recommendations