Classical probability model for Bell inequality

TL;DR

This paper demonstrates that by considering the randomness of experimental contexts, one can construct a common classical probability space for Bell test data, challenging the notion that Bell inequality violations imply non-classicality.

Contribution

It introduces the concept of 'Kolmogorovization' of contextuality, showing that Bell inequality violations can be explained within a classical probabilistic framework when context randomness is included.

Findings

Bell inequality violations can be modeled classically with context randomness.

A common probability space can be constructed for incompatible quantum experiments.

Classical probabilistic models can describe polarization measurements in quantum optics.

Abstract

We show that by taking into account randomness of realization of experimental contexts it is possible to construct common Kolmogorov space for data collected for these contexts, although they can be incompatible. We call such a construction "Kolmogorovization" of contextuality. This construction of common probability space is applied to Bell's inequality. It is well known that its violation is a consequence of collecting statistical data in a few incompatible experiments. In experiments performed in quantum optics contexts are determined by selections of pairs of angles $(\theta_i, \theta^\prime_j)$ fixing orientations of polarization beam splitters. Opposite to the common opinion, we show that statistical data corresponding to measurements of polarizations of photons in the singlet state, e.g., in the form of correlations, can be described in the classical probabilistic framework. The crucial point is that in constructing the common probability space one has to take into account not only randomness of the source (as Bell did), but also randomness of context-realizations (in particular, realizations of pairs of angles $(\theta_i, \theta^\prime_j)$). One may (but need not) say that randomness of "free will" has to be accounted.