Remarks on the use of objective probabilities in Bell-CHSH inequalities

TL;DR

This paper examines how different objective probability theories, like frequentism and propensities, influence the interpretation of Bell-CHSH inequalities and their violations, questioning the role of probability spaces and counterfactuals.

Contribution

It analyzes the implications of frequentist and propensity-based probabilities on Bell inequalities, highlighting how these frameworks affect the interpretation of quantum nonlocality and contextuality.

Findings

Frequentism rejects counterfactuals in Bell tests.

Propensities allow counterfactuals but have limited practical use.

Violations of Bell inequalities may stem from probability assumptions.

Abstract

The violation of Bell inequalities is often interpreted as showing that, if hidden variables exist, they must be contextual and non local. But they can also be explained questioning the probability space employed, or the validity of the Kolmogorov axioms. In this article we explore the additional constrains which can be deduced from two widely used objetive probability theories: frequentism and propensities. One of the strongest objections in the deduction of one version of Bell inequalities goes about the probability space, which assumes the existence of values for the output of the experiment in each run, while only two of the four values can be measured each time, making them counterfactual. It is shown that frequentism rejects the possibility of using counterfactual situations, while long-run propensities allow their use. In this case the introduction of locality and contextuality does not help to explain the violation, and an alternative explanation could point to a failure of the probability. Single case propensities were designed to associate probabilities to single events, but they need to be conditional to the whole universe, and do not have a clear link with the observed relative frequencies. It heavily limits their use.