Contrasting Implications of the Frequentist and Bayesian Interpretations of Probability when Applied to Quantum Mechanics Theory

TL;DR

This paper compares the implications of frequentist and Bayesian interpretations of probability in quantum mechanics, revealing how each affects notions of non-locality, hidden variables, and the validity of Bell inequalities.

Contribution

It offers a novel analysis of quantum probabilities through the lens of Bayesian interpretation, challenging traditional views on non-locality and hidden variables in quantum theory.

Findings

Frequentist interpretation implies non-locality in QM spin distributions.

Bayesian interpretation removes the need for non-locality assumptions.

A new stochastic hidden-variable model consistent with Bayesian probabilities is proposed.

Abstract

An examination is made of the differing implications from applying the two mainstream interpretations of probability, frequentist and Bayesian, to QM (quantum mechanics) theory for the Bohm-EPR experiment. The joint probability distribution for the possible spin outcomes for two particles with coupled spins is examined. Contrasting conclusions are made: (i) Under the frequentist interpretation, the QM spin distribution implies a widely-discussed non-locality because probabilistic conditioning on a spin measurement is viewed as corresponding to a causal influence. Under the Bayesian interpretation, this conditioning is viewed as providing information relevant to the spin probabilities and the argument for non-locality loses its force. (ii) The frequentist interpretation leads to the locality condition used by John Bell in 1964 to establish conditions for the existence of hidden variables behind the spin probability distribution but this condition is not consistent with the product (Bayes) rule of probability theory. Under the Bayesian interpretation, there is no motivation for this locality condition and a new stochastic hidden-variable model is given that reproduces the spin distribution. (iii) Bell's original 1964 inequality provides a necessary condition for the existence of a third-order joint probability distribution for three spin variables that is compatible through marginalization with the second-order joint distributions for the three possible spin pairs. Bell's inequality is logically equivalent to the 1969 CHSH inequality that involves expectations of four pairs of four spin variables; the CHSH inequality must be satisfied in order for a fourth-order joint probability distribution for the four spin variables to exist. When any of these inequalities are violated, a joint probability distribution for three spin outcomes fails to be valid because some probabilities are negative.