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Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

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Abstract

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.

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References

  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)

    Article  ADS  Google Scholar 

  2. Acin A., Masanes L., Gisin N.: From Bell’s Theorem to Secure Quantum Key Distribution. Phys. Rev. Lett. 97, 120405 (2006)

    Article  ADS  Google Scholar 

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

    Google Scholar 

  4. Bennett C., Sharpley R.: Interpolation of operators. Academic Press, London-New York (1988)

    MATH  Google Scholar 

  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, 2008

  6. Brassard G., Broadbent A., Tapp A.: Quantum Pseudo-Telepathy. Found. Phys. 35(11), 1877–1907 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Briët, J., Buhrman, H., Toner, B.: A generalized Grothendieck inequality and entanglement in XOR games. http://arXiv.org/abs/0901.2009v1 [quant-ph], 2009

  8. Brunner N., Gisin N., Scarani V., Simon C.: Detection loophole in asymmetric Bell experiments. Phys. Rev. Lett. 98, 220403 (2007)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  10. Buhrman H., Cleve R., Massar S., de Wolf R.: Non-locality and Communication Complexity. Rev. Mod. Phys. 82, 665 (2010)

    Article  ADS  Google Scholar 

  11. Cabello A., Larsson J.-A.: Minimum detection efficiency for a loophole-free atom-photon Bell experiment. Phys. Rev. Lett. 98, 220402 (2007)

    Article  ADS  Google Scholar 

  12. Cabello A., Rodriguez D., Villanueva I.: Necessary and sufficient detection efficiency for the Mermin inequalities. Rev. Lett. 101, 120402 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  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. 236–249

  14. Cleve, R., Gavinsly, D., Jain, R.: Entanglement-Resistant Two-Prover Interactive Proof Systems and Non-Adaptive Private Information Retrieval Systems. http://arXiv.org/abs/quant-ph/0707.1729v1 [quant-ph], 2007

  15. Cohen A., Dahmen W., DeVore R.: Compressed sensing and best k-term approximation. JAMS 22(1), 211–231 (2009)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    MATH  Google Scholar 

  18. Degorre J., Kaplan M., Laplante S., Roland J.: The communication complexity of non-signaling distributions. Lect. Notes Comp. Sci. 5734, 270–281 (2009)

    Article  ADS  Google Scholar 

  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. 199–210

  20. Effros E.G., Ruan Z.-J.: Operator spaces. London Math. Soc. Monographs New Series Oxford, Clarendon Press (2000)

    Google Scholar 

  21. Einstein A., Podolsky B., Rosen N.: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?. Phys. Rev. 47, 777 (1935)

    Article  MATH  ADS  Google Scholar 

  22. Grothendieck A.: Résumé de la théorie métrique des produits tensoriels topologiques (French). Bol. Soc. Mat. São Paulo 8, 1–79 (1953)

    MathSciNet  Google Scholar 

  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, 2007

  24. Jain, R., Ji, Z., Upadhyay, S., Watrous, J.: QIP = PSPACE. http://arXiv.org./abs/0907.4737v2 [quant-ph], 2009

  25. Junge, M.: Factorization theory for Spaces of Operators. Habilitationsschrift Kiel, 1996; see also: http://www.math.uiuc.edu/~mjunge/publish.html

  26. Junge M., Parcet J.: Rosenthal’s theorem for subspaces of noncommutative Lp. Duke Math. J. 141, 75–122 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Junge, M., Parcet, J.: Mixed-norm inequalities and operator space Lp embedding theory. Mem. Amer. Math. Soc. 952 (2010)

  28. Junge M., Parcet J.: A transference method in quantum probability. Adv. Math. 225, 389–444 (2010)

    Article  MATH  Google Scholar 

  29. Junge M., Parcet J., Xu Q.: Rosenthal type inequalities for free chaos. Ann. Probab. 35, 1374–1437 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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, 2008

  31. Kempe, J., Kobayashi, H., Matsumoto, K., Toner, B., Vidick, T.: Entangled games are hard to approximate. http://arXiv.org/abs/0704.2903v2 [quant-ph], 2007

  32. Khot, S., Vishnoi, N.K.: The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1. In: Proc. 46th IEEE Symp. on Foundations of Computer Science, Los Alamitos, CA: IEEE, 2005, pp. 53–62

  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)

    Article  ADS  Google Scholar 

  34. Ledoux M., Talagrand M.: Probability in Banach Spaces. Springer-Verlag, Berlin-Heidelberg-New York (1991)

    MATH  Google Scholar 

  35. Marcus, M.B., Pisier, G.: Random Fourier series with applications to Harmonic Analysis. Annals of Math. Studies, 101, Princeton, NJ: Princeton Univ. Press, 1981

  36. Masanes, Ll., Renner, R., Winter, A., Barrett, J., Christandl, M.: Security of key distribution from causality constraints. http://arXiv.org/abs/quant-ph/0606049v4 (2006)

  37. Masanes L.: Universally-composable privacy amplification from causality constraints. Phys. Rev. Lett. 102, 140501 (2009)

    Article  ADS  Google Scholar 

  38. Massar S.: Nonlocality, closing the detection loophole, and communication complexity. Phys. Rev. A 65, 032121 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  39. Massar S., Pironio S.: Violation of local realism vs detection efficiency. Phys. Rev. A 68, 062109 (2003)

    Article  ADS  Google Scholar 

  40. Paulsen, V.I.: Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics 78, Cambridge: Cambridge University Press, 2003

  41. Pearle P.M.: Hidden-variable example based upon data rejection. Phys. Rev. D 2, 1418 (1970)

    Article  ADS  Google Scholar 

  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)

    Article  MATH  ADS  Google Scholar 

  43. Pisier, G.: An Introduction to Operator Spaces. London Math. Soc. Lecture Notes Series 294, Cambridge: Cambridge University Press, 2003

  44. Pisier G.: The volume of convex bodies and Banach Space Geometry. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  45. Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS 60, Providence, RI: Amer. Math. Soc., 1986

  46. Pitowsky I.: New Bell inequalities for the singlet state: Going beyond the Grothendieck bound. J. Math. Phys. 49, 012101 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  47. Rao, A.: Parallel repetition in projection games and a concentration bound. In: 40th STOC Proc, STOC2008, New York: Assoc. for Computing Machinery, 2008

  48. Raz R.: A Parallel Repetition Theorem. SIAM J. Comp. 27, 763–803 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  49. Shor P.W., Preskill J.: Simple Proof of Security of the BB84 Quantum Key Distribution Protocol. Phys. Rev. Lett. 85, 441–444 (2000)

    Article  ADS  Google Scholar 

  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, 1989

  51. Tsirelson B.S.: Hadronic Journal Supplement 84, 329–345 (1993)

    MathSciNet  Google Scholar 

  52. Vertesi T., Pal K.F.: Bounding the dimension of bipartite quantum systems. Phys. Rev. A 79, 042106 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  54. Werner R.F., Wolf M.M.: Bell inequalities and Entanglement. Quant. Inf. Comp. 1(3), 1–25 (2001)

    MATH  MathSciNet  Google Scholar 

  55. Wolf M.M., Pérez-García D.: Assessing dimensions from evolution. Phys. Rev. Lett. 102, 190504 (2009)

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Illinois, 61801-2975, USA

    M. Junge & C. Palazuelos

  2. Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040, Madrid, Spain

    C. Palazuelos, D. Pérez-García & I. Villanueva

  3. Niels Bohr Institute, 2100, Copenhagen, Denmark

    M. M. Wolf

Authors
  1. M. Junge
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  2. C. Palazuelos
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  3. D. Pérez-García
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  4. I. Villanueva
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  5. M. M. Wolf
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Correspondence to D. Pérez-García.

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Communicated by M.B. Ruskai

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Junge, M., Palazuelos, C., Pérez-García, D. et al. Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory. Commun. Math. Phys. 300, 715–739 (2010). https://doi.org/10.1007/s00220-010-1125-5

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