Foundations of Physics

, Volume 45, Issue 7, pp 711–725 | Cite as

CHSH Inequality: Quantum Probabilities as Classical Conditional Probabilities

  • Andrei Khrennikov


In this note we demonstrate that the results of observations in the EPR–Bohm–Bell experiment can be described within the classical probabilistic framework. However, the “quantum probabilities” have to be interpreted as conditional probabilities, where conditioning is with respect to fixed experimental settings. Our approach is based on the complete account of randomness involved in the experiment. The crucial point is that randomness of selections of experimental settings has to be taken into account within one consistent framework covering all events related to the experiment. This approach can be applied to any complex experiment in which statistical data are collected for various (in general incompatible) experimental settings.


Classical and quantum probability Bell inequality  CHSH inequality Randomness of experimental settings Conditional probabilities No signaling Kolmogorov probability model 



This paper was written during author’s visiting professor fellowship to the Institute for Quantum Optics and Quantum Information of Austrian Academy of Science (April–June, 2014); the main result of this paper was presented in the course of lectures on the inter-relation between classical and quantum randomness given for the graduate students of this institute. I would like to thank Anton Zeilinger for hospitality and critical discussions about the objective representation of quantum observables.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.International Center for Mathematical Modeling in Physics and Cognitive SciencesLinnaeus UniversityVäxjö-KalmarSweden

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