Communications in Mathematical Physics

, Volume 300, Issue 3, pp 715–739 | Cite as

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

  • M. Junge
  • C. Palazuelos
  • D. Pérez-García
  • I. Villanueva
  • M. M. Wolf


In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order \({{\rm \Omega} \left(\frac{\sqrt{n}}{\log^2n} \right)}\) when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative L p embedding theory.

As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise.


Operator Space Bell Inequality Tensor Norm Complete Contraction Parallel Repetition 
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.
    Acin A., Brunner N., Gisin N., Massar S., Pironio S., Scarani V.: Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007)CrossRefADSGoogle Scholar
  2. 2.
    Acin A., Masanes L., Gisin N.: From Bell’s Theorem to Secure Quantum Key Distribution. Phys. Rev. Lett. 97, 120405 (2006)CrossRefADSGoogle Scholar
  3. 3.
    Bell J.S.: On the Einstein-Poldolsky-Rosen paradox. Physics 1, 195 (1964)Google Scholar
  4. 4.
    Bennett C., Sharpley R.: Interpolation of operators. Academic Press, London-New York (1988)zbMATHGoogle Scholar
  5. 5.
    Ben-Or, M., Hassidim, A., Pilpel, H.: Quantum Multi Prover Interactive Proofs with Communicating Provers. In: Proceedings of 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), Los Alamitos, CA: IEEE, 2008Google Scholar
  6. 6.
    Brassard G., Broadbent A., Tapp A.: Quantum Pseudo-Telepathy. Found. Phys. 35(11), 1877–1907 (2005)zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Briët, J., Buhrman, H., Toner, B.: A generalized Grothendieck inequality and entanglement in XOR games. [quant-ph], 2009
  8. 8.
    Brunner N., Gisin N., Scarani V., Simon C.: Detection loophole in asymmetric Bell experiments. Phys. Rev. Lett. 98, 220403 (2007)CrossRefADSGoogle Scholar
  9. 9.
    Brunner N., Pironio S., Acin A., Gisin N., Methot A.A., Scarani V.: Testing the Hilbert space dimension. Phys. Rev. Lett. 100, 210503 (2008)CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Buhrman H., Cleve R., Massar S., de Wolf R.: Non-locality and Communication Complexity. Rev. Mod. Phys. 82, 665 (2010)CrossRefADSGoogle Scholar
  11. 11.
    Cabello A., Larsson J.-A.: Minimum detection efficiency for a loophole-free atom-photon Bell experiment. Phys. Rev. Lett. 98, 220402 (2007)CrossRefADSGoogle Scholar
  12. 12.
    Cabello A., Rodriguez D., Villanueva I.: Necessary and sufficient detection efficiency for the Mermin inequalities. Rev. Lett. 101, 120402 (2008)CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Cleve, R., Høyer, P., Toner, B., Watrous, J.: Consequences and Limits of Nonlocal Strategies. In: Proceedings of the 19th IEEE Annual Conference on Computational Complexity (CCC 2004), Los Alamitos, CA: IEEE, pp. 236–249Google Scholar
  14. 14.
    Cleve, R., Gavinsly, D., Jain, R.: Entanglement-Resistant Two-Prover Interactive Proof Systems and Non-Adaptive Private Information Retrieval Systems. [quant-ph], 2007
  15. 15.
    Cohen A., Dahmen W., DeVore R.: Compressed sensing and best k-term approximation. JAMS 22(1), 211–231 (2009)MathSciNetGoogle Scholar
  16. 16.
    Collins D., Gisin N.: A relevant two qubit Bell inequality inequivalent to the CHSH inequality. J. Phys. A: Math. Gen. 37, 1775–1787 (2004)zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Defant A., Floret K.: Tensor Norms and Operator Ideals. North-Holland, Amsterdam (1993)zbMATHGoogle Scholar
  18. 18.
    Degorre J., Kaplan M., Laplante S., Roland J.: The communication complexity of non-signaling distributions. Lect. Notes Comp. Sci. 5734, 270–281 (2009)CrossRefADSGoogle Scholar
  19. 19.
    Doherty, A.C., Liang, Y-C., Toner, B., Wehner, S.: The quantum moment problem and bounds on entangled multi-prover games. In: Proc. of IEEE Conference on Computational Complexity 2008, Los Alamitos, CA: IEEE, pp. 199–210Google Scholar
  20. 20.
    Effros E.G., Ruan Z.-J.: Operator spaces. London Math. Soc. Monographs New Series Oxford, Clarendon Press (2000)Google Scholar
  21. 21.
    Einstein A., Podolsky B., Rosen N.: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?. Phys. Rev. 47, 777 (1935)zbMATHCrossRefADSGoogle Scholar
  22. 22.
    Grothendieck A.: Résumé de la théorie métrique des produits tensoriels topologiques (French). Bol. Soc. Mat. São Paulo 8, 1–79 (1953)MathSciNetGoogle Scholar
  23. 23.
    Holenstein, T.: Parallel repetition: simplifications and the no-signaling case. In: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing STOC 2007, New York: Assoc. for Computing Machinery, 2007Google Scholar
  24. 24.
    Jain, R., Ji, Z., Upadhyay, S., Watrous, J.: QIP = PSPACE. [quant-ph], 2009
  25. 25.
    Junge, M.: Factorization theory for Spaces of Operators. Habilitationsschrift Kiel, 1996; see also:
  26. 26.
    Junge M., Parcet J.: Rosenthal’s theorem for subspaces of noncommutative Lp. Duke Math. J. 141, 75–122 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Junge, M., Parcet, J.: Mixed-norm inequalities and operator space Lp embedding theory. Mem. Amer. Math. Soc. 952 (2010)Google Scholar
  28. 28.
    Junge M., Parcet J.: A transference method in quantum probability. Adv. Math. 225, 389–444 (2010)zbMATHCrossRefGoogle Scholar
  29. 29.
    Junge M., Parcet J., Xu Q.: Rosenthal type inequalities for free chaos. Ann. Probab. 35, 1374–1437 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Kempe, J., Regev, O., Toner, B.: The Unique Games Conjecture with Entangled Provers is False. In: Proceedings of 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), Los Alamitos, CA: IEEE, 2008Google Scholar
  31. 31.
    Kempe, J., Kobayashi, H., Matsumoto, K., Toner, B., Vidick, T.: Entangled games are hard to approximate. [quant-ph], 2007
  32. 32.
    Khot, S., Vishnoi, N.K.: The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into1. In: Proc. 46th IEEE Symp. on Foundations of Computer Science, Los Alamitos, CA: IEEE, 2005, pp. 53–62Google Scholar
  33. 33.
    Kraus , B. , Gisin , N. , Renner R.: Lower and upper bounds on the secret key rate for QKD protocols using one–way classical communication. Phys. Rev. Lett. 95, 080501 (2005)CrossRefADSGoogle Scholar
  34. 34.
    Ledoux M., Talagrand M.: Probability in Banach Spaces. Springer-Verlag, Berlin-Heidelberg-New York (1991)zbMATHGoogle Scholar
  35. 35.
    Marcus, M.B., Pisier, G.: Random Fourier series with applications to Harmonic Analysis. Annals of Math. Studies, 101, Princeton, NJ: Princeton Univ. Press, 1981Google Scholar
  36. 36.
    Masanes, Ll., Renner, R., Winter, A., Barrett, J., Christandl, M.: Security of key distribution from causality constraints. (2006)
  37. 37.
    Masanes L.: Universally-composable privacy amplification from causality constraints. Phys. Rev. Lett. 102, 140501 (2009)CrossRefADSGoogle Scholar
  38. 38.
    Massar S.: Nonlocality, closing the detection loophole, and communication complexity. Phys. Rev. A 65, 032121 (2002)CrossRefMathSciNetADSGoogle Scholar
  39. 39.
    Massar S., Pironio S.: Violation of local realism vs detection efficiency. Phys. Rev. A 68, 062109 (2003)CrossRefADSGoogle Scholar
  40. 40.
    Paulsen, V.I.: Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics 78, Cambridge: Cambridge University Press, 2003Google Scholar
  41. 41.
    Pearle P.M.: Hidden-variable example based upon data rejection. Phys. Rev. D 2, 1418 (1970)CrossRefADSGoogle Scholar
  42. 42.
    Pérez-García D., Wolf M.M., Palazuelos C., Villanueva I., Junge M.: Unbounded violation of tripartite Bell inequalities. Commun. Math. Phys. 279(2), 455–486 (2008)zbMATHCrossRefADSGoogle Scholar
  43. 43.
    Pisier, G.: An Introduction to Operator Spaces. London Math. Soc. Lecture Notes Series 294, Cambridge: Cambridge University Press, 2003Google Scholar
  44. 44.
    Pisier G.: The volume of convex bodies and Banach Space Geometry. Cambridge University Press, Cambridge (1989)zbMATHCrossRefGoogle Scholar
  45. 45.
    Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS 60, Providence, RI: Amer. Math. Soc., 1986Google Scholar
  46. 46.
    Pitowsky I.: New Bell inequalities for the singlet state: Going beyond the Grothendieck bound. J. Math. Phys. 49, 012101 (2008)CrossRefMathSciNetADSGoogle Scholar
  47. 47.
    Rao, A.: Parallel repetition in projection games and a concentration bound. In: 40th STOC Proc, STOC2008, New York: Assoc. for Computing Machinery, 2008Google Scholar
  48. 48.
    Raz R.: A Parallel Repetition Theorem. SIAM J. Comp. 27, 763–803 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Shor P.W., Preskill J.: Simple Proof of Security of the BB84 Quantum Key Distribution Protocol. Phys. Rev. Lett. 85, 441–444 (2000)CrossRefADSGoogle Scholar
  50. 50.
    Tomczak-Jaegermann, N.: Banach-Mazur Distances and Finite Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics 38, London: Longman Scientific and Technical, 1989Google Scholar
  51. 51.
    Tsirelson B.S.: Hadronic Journal Supplement 84, 329–345 (1993)MathSciNetGoogle Scholar
  52. 52.
    Vertesi T., Pal K.F.: Bounding the dimension of bipartite quantum systems. Phys. Rev. A 79, 042106 (2009)CrossRefMathSciNetADSGoogle Scholar
  53. 53.
    Wehner S., Christandl M., Doherty A.C.: A lower bound on the dimension of a quantum system given measured data. Phys. Rev. A 78, 062112 (2008)CrossRefADSGoogle Scholar
  54. 54.
    Werner R.F., Wolf M.M.: Bell inequalities and Entanglement. Quant. Inf. Comp. 1(3), 1–25 (2001)zbMATHMathSciNetGoogle Scholar
  55. 55.
    Wolf M.M., Pérez-García D.: Assessing dimensions from evolution. Phys. Rev. Lett. 102, 190504 (2009)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • M. Junge
    • 1
  • C. Palazuelos
    • 1
    • 2
  • D. Pérez-García
    • 2
  • I. Villanueva
    • 2
  • M. M. Wolf
    • 3
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignIllinoisUSA
  2. 2.Departamento de Análisis MatemáticoUniversidad Complutense de MadridMadridSpain
  3. 3.Niels Bohr InstituteCopenhagenDenmark

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