Skip to main content
SpringerLink
Log in

Random Constructions in Bell Inequalities: A Survey

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Initially motivated by their relevance in foundations of quantum mechanics and more recently by their applications in different contexts of quantum information science, violations of Bell inequalities have been extensively studied during the last years. In particular, an important effort has been made in order to quantify such Bell violations. Probabilistic techniques have been heavily used in this context with two different purposes. First, to quantify how common the phenomenon of Bell violations is; and second, to find large Bell violations in order to better understand the possibilities and limitations of this phenomenon. However, the strong mathematical content of these results has discouraged some of the potentially interested readers. The aim of the present work is to review some of the recent results in this direction by focusing on the main ideas and removing most of the technical details, to make the previous study more accessible to a wide audience.

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. Formally, Bell talked about a local hidden variable model (LHVM).

  2. Note that P is not a probability distribution itself. For every fixed x, y we have that \((P(a,b|x,y))_{a,b=1}^K\) is a probability distribution. However, it is standard to use this terminology. Sometimes they are also called behaviors, because of the terminology used in [49].

  3. Note that any Bell functional M defines an inequality by writing \(\langle M, P\rangle \le \omega (M)\), \(P\in {\mathcal {L}}\).

  4. The subscript 2 in \({\mathcal {D}}_2(N)\) refers to the bipartite case.

  5. The exact value of the Grothendieck constant is still unknown in both the real and the complex case (see [12] for the most recent progress).

  6. The work [3] deals with bipartite XOR games, but the problem is completely equivalent.

  7. In fact, the modified Marcenko–Pastur law proved in [3, Theorem 3] gives the probability for each of these vectors to have norm larger than \( \sqrt{f(2-\epsilon )+\delta }\). Then, the authors proved that Eq. (1.3) is not affected if one rules out those vectors with large norm.

  8. In fact, we state here a slightly modified version for T real, which involves a modification in the constant (see [14, Theorem 9] for details).

  9. The result can be generalized to n parties straightforwardly.

  10. One needs to normalize the gaussian state to have the same distribution.

  11. Pisier has shown in [41] that such a factor can be reduced to \(\log ^{-\frac{3}{2}} N\).

  12. The only property used by the authors is that the tensor products of j Pauli matrices form an orthogonal basis of \(M_N\) formed by observables. Any other such a system would equally work in the proof.

  13. We will not talk about symmetric correlation Bell functionals since these should be defined as those functionals which are invariant under permutations of the parties without any extra restriction.

  14. Here, we really mean the equations defining the facets of the set \({\mathcal {L}}\).

References

  1. Ambainis, A., Iraids, J.: provable advantage for quantum strategies in random symmetric XOR games. arXiv:1302.2347

  2. Ambainis, A., Kravchenko, D., Nahimovs, N., Rivosh, A.: Nonlocal quantum XOR games for large number of players. In: Theory and Applications of Models of Computation. Lecture Notes in Computer Science, vol. 6108, pp. 72–83. Springer, Berlin (2010)

  3. Ambainis, A., Backurs, A., Balodis, K., Kravcenko, D., Ozols, R., Smotrovs, J., Virza, M.: Quantum strategies are better than classical in almost any XOR game. In: Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 7391, pp. 25–37. Springer, Heidelberg (2012)

  4. Ambainis, A., Iraids, J., Kravchenko, D., Virza, M.: Advantage of quantum strategies in random symmetric XOR games. In: Mathematical and Engineering Methods in Computer Science. Lecture Notes in Computer Science, vol. 7721, pp. 57–68. Springer, Berlin (2013)

  5. Ardehali, M.: Bell inequalities with a magnitude of violation that grows exponentially with the number of particles. Phys. Rev. A 46(9), 5375–5378 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  6. Atkin, M.R., Zohren, S.: Violations of Bell inequalities from random pure states. arXiv:1407.8233

  7. Aubrun, G., Szarek, S., Ye, D.: Entanglement thresholds for random induced states. Commun. Pure Appl. Math. 67, 129–171 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  9. Blei, R.C.: Multidimensional extensions of the Grothendieck inequality and applications. Arkiv fur Matematik 17, 51–68 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Bombal, F., Pérez-García, D., Villanueva, I.: Multilinear extensions of Grothendieck’s theorem. Q. J. Math. 55, 441–450 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Braverman, M., Makarychev, K., Makarychev, Y., Naor, A.: The Grothendieck constant is strictly smaller than Krivine’s bound. Forum Math. Pi 1, e4, 42 (2013)

  13. 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 

  14. Briët, J., Buhrman, H., Lee, T., Vidick, T.: Multipartite entanglement in XOR games. Quant. Inf. Process. 13, 334–360 (2013)

    MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  17. 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)

  18. Carne, T.K.: Banach lattices and extensions of Grothendieck’s inequality. J. Lond. Math. Soc. 21(3), 496–516 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed Experiment to test local-hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969)

    Article  ADS  MATH  Google Scholar 

  20. Collins, D., Gisin, N., Linden, N., Massar, S., Popescu, S.: Bell inequalities for arbitrarily high dimensional systems. Phys. Rev. Lett. 88(4), 040404 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. Drumond, R.C., Oliveira, R.I.: Small violations of full correlations Bell inequalities for multipartite pure random states. Phys. Rev. A 86(1), 012117 (2012)

    Article  ADS  Google Scholar 

  23. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)

    Article  ADS  MATH  Google Scholar 

  24. González-Guillén, C.E., Jiménez, C.H., Palazuelos, C., Villanueva, I.: Sampling quantum nonlocal correlations with high probability. Commun. Math. Phys. 344, 141–154 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. González-Guillén, C.E., Lancien, C., Palazuelos, C., Villanueva, I.: Random quantum correlations are generically non-classical. Ann. Henri Poincaré. 18, 3793–3813 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. González-Guillén, C., Palazuelos, C., Villanueva, I.: Distance between Haar unitary and random Gaussian matrices. J. Theor. Probab. https://doi.org/10.1007/s10959-016-0712-6

  27. 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 

  28. Junge, M., Palazuelos, C., Pérez-García, D., Villanueva, I., Wolf, M.M.: Unbounded violations of bipartite Bell inequalities via operator space theory. Commun. Math. Phys. 300(3), 715–739 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Junge, M., Palazuelos, C., Pérez-García, D., Villanueva, I., Wolf, M.M.: Operator space theory: a natural framework for Bell inequalities. Phys. Rev. Lett. 104, 170405 (2010)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. 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)

  32. Khot, S., Vishnoi, N.: The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into \(\ell _1\). In: Proceedings of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005)

  33. Liang, Y.-C., Harrigan, N., Bartlett, S.D., Rudolph, T.: Nonclassical correlations from randomly chosen local measurements. Phys. Rev. Lett. 104, 050401 (2010)

    Article  ADS  Google Scholar 

  34. Liang, Y.-C., Vertesi, T., Brunner, N.: Semi-device-independent bounds on entanglement. Phys. Rev. A 83, 022108 (2011)

    Article  ADS  Google Scholar 

  35. Marcenko, V.A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Math. USSR Sbornik 1, 457–483 (1967)

    Article  Google Scholar 

  36. Mermin, D.: Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65(15), 1838–1840 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Palazuelos, C.: On the largest Bell violation attainable by a quantum state. J. Funct. Anal. 267(7), 1959–1985 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. 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  ADS  MathSciNet  MATH  Google Scholar 

  40. Pisier, G.: Grothendieck’s Theorem, past and present. Bull. Am. Math. Soc. 49, 237–323 (2012). See also arXiv:1101.4195

  41. Pisier, G.: Tripartite Bell inequality, random matrices and trilinear forms. arXiv:1203.2509

  42. Pitowsky, I.: Macroscopic objects in quantum mechanics: a combinatorial approach. Phys. Rev. A 70, 022103-1-6 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Raz, R.: A parallel repetition theorem. SIAM J. Comput. 27, 763–803 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  44. Regev, O.: Bell violations through independent bases games. Quant. Inf. Comput. 12(1–2), 9–20 (2012)

    MathSciNet  MATH  Google Scholar 

  45. Salem, R., Zygmund, A.: Some properties of trigonometric series whose terms have random signs. Acta Math. 91(1), 245–301 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shadbolt, P., Vertesi, T., Liang, Y.-C., Branciard, C., Brunner, N., O’Brien, J.: Guaranteed violation of a Bell inequality without aligned reference frames or calibrated devices. Sci. Rep. 2, 470 (2012)

    Article  Google Scholar 

  47. Szarek, S., Werner, E., Zyczkowski, K.: How often is a random quantum state \(k\)-entangled? J. Phys. A 44, 045303 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Tonge, A.: The Von Neumann inequality for polynomials in several Hilbert–Schmidt operators. J. Lond. Math. 2(1), 519–526 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  50. Vidick, T., Wehner, S.: More nonlocality with less entanglement. Phys. Rev. A 83, 052310 (2011)

    Article  ADS  Google Scholar 

  51. Wallman, J.J., Liang, Y.-C., Bartlett, S.D.: Generating nonclassical correlations without fully aligning measurements. Phys. Rev. A 83, 022110 (2011)

    Article  ADS  Google Scholar 

  52. Werner, R.F., Wolf, M.M.: Bell inequalities and entanglement. Quant. Inf. Comput. 1(3), 1–25 (2001)

    MathSciNet  MATH  Google Scholar 

  53. Werner, R.F., Wolf, M.M.: All multipartite Bell correlation inequalities for two dichotomic observables per site. Phys. Rev. A 64, 032112 (2001)

    Article  ADS  Google Scholar 

  54. Zukowski, M., Brukner, C.: Bell’s theorem for general \(N\)-qubit states. Phys. Rev. Lett. 88, 210401 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Zyczkowski, K., Penson, K.A., Nechita, I., Collins, B.: Generating random density matrices. J. Math. Phys. 52(6), 062201 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the referees for their many useful comments which helped to improve the readability of this paper. Author’s research was supported by the Spanish projects MTM2011-26912, Comunidad de Madrid QUITEMAD+ S2013/ICE-2801, and MINECO: ICMAT Severo Ochoa Project SEV-2011-0087 and the “Ramón y Cajal” program (RYC-2012-10449).

Author information

Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (ICMAT), Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Carlos Palazuelos

Authors
  1. Carlos Palazuelos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Palazuelos.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Palazuelos, C. Random Constructions in Bell Inequalities: A Survey. Found Phys 48, 857–885 (2018). https://doi.org/10.1007/s10701-017-0135-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-017-0135-y

Keywords

Search

Navigation