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


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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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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