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Random Quantum Correlations are Generically Non-classical

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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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Authors and Affiliations

  1. Departamento de Matemática Aplicada a la Ingeniería Industrial E.T.S.I. Industriales, Universidad Politécnica de Madrid, 28006, Madrid, Spain

    Carlos E. González-Guillén

  2. Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Carlos E. González-Guillén & Ignacio Villanueva

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

    Cécilia Lancien, Carlos Palazuelos & Ignacio Villanueva

  4. Instituto de Ciencias Matemáticas, 28049, Madrid, Spain

    Cécilia Lancien & Carlos Palazuelos

Authors
  1. Carlos E. González-Guillén
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  2. Cécilia Lancien
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  3. Carlos Palazuelos
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  4. Ignacio Villanueva
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Corresponding author

Correspondence to Carlos E. González-Guillén.

Additional information

Communicated by David Pérez-García.

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González-Guillén, C.E., Lancien, C., Palazuelos, C. et al. Random Quantum Correlations are Generically Non-classical. Ann. Henri Poincaré 18, 3793–3813 (2017). https://doi.org/10.1007/s00023-017-0615-9

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  • DOI: https://doi.org/10.1007/s00023-017-0615-9

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