The CHSH Test Can be Used for All Quantum-Describable Systems


This paper shows that the ClauserHorneShimonyHolt test of locality of correlations which was originally designed to be used with binary observables can actually be used for any couples of quantum-like bounded continuous observables, and then for any experimental situation describable within the mathematical framework of quantum theory.

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  1. 1.

    For discrete spectrum observables, the integral symbol is replaced by a discrete sum over the corresponding λα and λβ.


  1. 1.

    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)

    ADS  Article  Google Scholar 

  2. 2.

    Aspect, A., et al.: Experimental test of Bell's inequalities using time-varying analyzers. Phys. Rev. Lett. 49(25), 1804–1807 (1982)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Tittel, W., Brendel, J., Gisin, B., Herzog, T., Zbinden, H., Gisin, N.: Experimental demonstration of quantum-correlations over more than 10 kilometers. Phys. Rev. A. 57(5), 3229–3232 (1998)

    ADS  Article  Google Scholar 

  4. 4.

    Lee, S.-W., Jeong, H., Jaksch, D.: Phys. Rev. A 80, 022104 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Wenger, J., Hafezi, M., Grosshans, F., Tualle-Brouri, R., Grangier, P.: Phys. Rev. A 67, 012105 (2003)

    ADS  Article  Google Scholar 

  6. 6.

    A. Ketterer, A. Keller, T. Coudreau and P. Milman. Testing the Clauser-Horne-Shimony-Holt inequality using observables with arbitrary spectrum, Phy Rev.91(1) (2005)

  7. 7.

    Gleason, A.: Measures on the Closed Subspaces of a Hilbert Space. J. Math. Mech. 6, 885–893 (1957)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Plancherel, M.: Contribution à l’étude de la representation d’une fonction arbitraire par les integrales définies. Rendiconti del Circol Matematico di Palermo 30, 298–335 (1910)

    MATH  Google Scholar 

  9. 9.

    Lamata, L., Léon, J.: Dealing with entanglement of continuous variables: Schmidt decomposition with discrete sets of orthogonal functions. J. Opt. B Quantum Semiclass. Opt. 7(2005), 224–229 (2005)

    ADS  Article  Google Scholar 

  10. 10.

    Uzan, P.: Deciding the Mind–body problem experimentally. Axiomathes 27(4), 333–354 (2017)

    Article  Google Scholar 

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Correspondence to Pierre Uzan.

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Uzan, P. The CHSH Test Can be Used for All Quantum-Describable Systems. Found Phys 50, 708–714 (2020).

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  • CHSH test
  • Non-locality
  • Entanglement
  • Quantum-like description