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Annales Henri Poincaré

, Volume 18, Issue 12, pp 3793–3813 | Cite as

Random Quantum Correlations are Generically Non-classical

  • Carlos E. González-GuillénEmail author
  • Cécilia Lancien
  • Carlos Palazuelos
  • Ignacio Villanueva
Article

Abstract

It is now a well-known fact that the correlations arising from local dichotomic measurements on an entangled quantum state may exhibit intrinsically non-classical features. In this paper we delve into a comprehensive study of random instances of such bipartite correlations. The main question we are interested in is: given a quantum correlation, taken at random, how likely is it that it is truly non-explainable by a classical model? We show that, under very general assumptions on the considered distribution, a random correlation which lies on the border of the quantum set is with high probability outside the classical set. What is more, we are able to provide the Bell inequality certifying this fact. On the technical side, our results follow from (i) estimating precisely the “quantum norm” of a random matrix and (ii) lower-bounding sharply enough its “classical norm”, hence proving a gap between the two. Along the way, we need a non-trivial upper bound on the \(\infty {\rightarrow }1\) norm of a random orthogonal matrix, which might be of independent interest.

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References

  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). arXiv:quant-ph/0702152 ADSCrossRefGoogle Scholar
  2. 2.
    Acin, A., Gisin, N., Massanes, L.: From Bell’s theorem to secure quantum key distribution. Phys. Rev. Lett. 97, 120405 (2006). arXiv:quant-ph/0510094 ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambainis, A., Bačkurs, A., Balodis, K., Kravčenko, D., Ozols, R., Smotrovs, J., Virza, M.: Quantum strategies are better than classical in almost any XOR games. In: Proceedings of the 39th ICALP, pp. 25–37 (2012). arXiv:1112.3330 [quant-ph]
  4. 4.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge (2010)Google Scholar
  5. 5.
    Aubrun, G., Szarek, S.J.: Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory. Mathematical Surveys and Monographs, vol. 223, American Mathematical Society (2017). http://math.univ-lyon1.fr/~aubrun/ABMB/index.html
  6. 6.
    Barvinok, A.: Measure Concentration. Math 710 Lecture Notes, Department of Mathematics, University of Michigan (2005). http://www.math.lsa.umich.edu/~barvinok/total710.pdf
  7. 7.
    Bell, J.S.: On the Einstein–Podolsky–Rosen paradox. Physics 1, 195–200 (1964)Google Scholar
  8. 8.
    Buhrman, H., Cleve, R., Massar, S., de Wolf, R.: Nonlocality and communication complexity. Rev. Mod. Phys. 82, 665 (2010). arXiv:0907.3584 [quant-ph]ADSCrossRefGoogle Scholar
  9. 9.
    Dupic, T., Pérez Castillo, I.: Spectral Density of Products of Wishart Dilute Random Matrices. Part I: The Dense Case. arXiv:1401.7802 [cond-mat.dis-nn]
  10. 10.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    González-Guillén, C.E., Jiménez, C.H., Palazuelos, C., Villanueva, I.: Sampling quantum nonlocal correlations with high probability. Comm. Math. Phys. 344(1), 141–154 (2016). arXiv:1412.4010 [quant-ph]ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Götze, F., Kösters, H., Tikhomirov, A.: Asymptotic spectra of matrix-valued functions of independent random matrices and free probability. Random Matrices: Theory Appl. 04, 1550005 (2015). arXiv:1408.1732 [math.PR]CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Hensen, B., Bernien, H., Dréau, A.E., Reiserer, A., Kalb, N., Blok, M.S., Ruitenberg, J., Vermeulen, R.F.L., Schouten, R.N., Abellán, C., Amaya, W., Pruneri, V., Mitchell, M.W., Markham, M., Twitchen, D.J., Elkouss, D., Wehner, S., Taminiau, T.H., Hanson, R.: Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature 526(7575), 682–686 (2015). arXiv:1508.05949 [quant-ph]ADSCrossRefGoogle Scholar
  14. 14.
    Levy, P.: Problèmes Concrets d’analyse Fonctionnelle, 2nd edn. Gauthier-Villars, Paris (1951). (in French)zbMATHGoogle Scholar
  15. 15.
    Meckes, E., Meckes, M.: Spectral measures of powers of random matrices. Electron. Commun. Probab. 18.78, 1–13 (2013). arXiv:1210.2681 [math.PR]zbMATHMathSciNetGoogle Scholar
  16. 16.
    Müller, R.R.: On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels. IEEE Trans. Inf. Theor. 48, 2086–2091 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Palazuelos, C.: Random Constructions in Bell Inequalities: A Survey. arXiv:1502.02175 [quant-ph]
  18. 18.
    Pironio, S., Acín, A., Massar, S., Boyer de la Giroday, A., Matsukevich, D.N., Maunz, P., Olmschenk, S., Hayes, D., Luo, L., Manning, T.A., Monroe, C.: Random numbers certified by Bell’s theorem. Nature 464, 1021 (2010). arXiv:0911.3427 [quant-ph]ADSCrossRefGoogle Scholar
  19. 19.
    Pisier, G.: Grothendieck’s theorem, past and present. Bull. Am. Math. Soc. 49, 237–323 (2012). arXiv:1101.4195 [math.FA]CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Tsirelson, B.S.: Some results and problems on quantum Bell-type inequalities. Hadron. J. Suppl. 8(4), 329–345 (1993)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Carlos E. González-Guillén
    • 1
    • 2
    Email author
  • Cécilia Lancien
    • 3
    • 4
  • Carlos Palazuelos
    • 3
    • 4
  • Ignacio Villanueva
    • 2
    • 3
  1. 1.Departamento de Matemática Aplicada a la Ingeniería Industrial E.T.S.I. IndustrialesUniversidad Politécnica de MadridMadridSpain
  2. 2.Instituto de Matemática InterdisciplinarUniversidad Complutense de MadridMadridSpain
  3. 3.Departamento de Análisis MatemáticoUniversidad Complutense de MadridMadridSpain
  4. 4.Instituto de Ciencias MatemáticasMadridSpain

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