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CHSH Inequality: Quantum Probabilities as Classical Conditional Probabilities


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.

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  1. Experiments to test violation of the Bell-type inequalities can be treated as one special class of statistical tests of nonclassicality of quantum probabilistic data. As was emphasized by Feynman et al. [16] and Accardi [17], another important test is given by the two slit experiment, see also [18]. However, recently the viewpoint that entanglement is the main root of quantumness became very popular in the quantum foundational community, especially in its part closely linked to quantum information and technology. (In particular, this viewpoint was presented in the private discussions of the author and Anton Zeilinger.) However, this question needs further clarifications and debates.

  2. The quantum formalism does not account this sort of randomness. Random generators used in the experimental tests based on the Bell-type inequalities are not described by operators in the complex Hilbert space. They are considered as “technicalities”; often this sort of randomness is related to the freewill of an experimentalist.

  3. As a possible realization, we can consider the following experimental framework. The random generator \(a\) is coupled to a block which splits the channel going from the source of photons in the \(A\)-direction into two channels, which are also labeled by \(i=1,2.\) Each of the channels is coupled to its own polarization beam splitter (PBS) which has the fixed orientation given by the angle \(\theta _i\) and the PBS in the \(i\)th channel is coupled to its pair of the detectors \(D^A_i(-),\) polarization down, and \(D^A_i(+),\) polarization up. Thus at each side there are two PBSs (corresponding to the fixed orientations) and totally 4 detectors. The complete two-side experimental scheme is based on 4 PBSs and 8 detectors.

  4. The terminology might be misleading. To escape this problem, we recall that in our model detectors have 100 % efficiency; nondetection means just that this experimental setting was not selected.


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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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Khrennikov, A. CHSH Inequality: Quantum Probabilities as Classical Conditional Probabilities. Found Phys 45, 711–725 (2015).

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