Test of a hypothesis of realism in quantum theory using a bayesian approach

TL;DR

This paper introduces new equality and inequality relations based on Bayesian probability for testing realism in quantum mechanics, which can be experimentally tested with fewer systems than traditional Bell or GHZ tests.

Contribution

The paper derives a novel equality and inequality for realism tests that require only two entangled systems, differing from traditional inequalities that need more complex setups.

Findings

The new equality can be tested with two entangled particles in a Bell state.

The inequality allows testing realism without the 'contextuality loophole'.

Violations are demonstrated with spin pairs and meson oscillations.

Abstract

In this paper we propose a time-independent \textit{equality} and time-dependent \textit{inequality}, suitable for an experimental test of the hypothesis of realism. The derivation of these relations is based on the concept of conditional probability and on Bayes' theorem in the framework of Kolmogorov's axiomatics of probability theory. The equality obtained is intrinsically different from the well known GHZ-equality and its variants, because violation of the new equality might be tested in experiments with only two microsystems in a maximally entangled Bell state $\big\vert\, \Psi^-\,\big\rangle$, while a test of the GHZ-equality requires at least three quantum systems in a special state $\big\vert\, \Psi^{\textrm{GHZ}}\,\big\rangle$. The obtained inequality differs from Bell's, Wigner's, and Leggett-Garg inequalities, because it deals with spin $s=1/2$ projections onto only two non-parallel directions at two different moments of time, while a test of the Bell and Wigner inequalities requires at least three non-parallel directions, and a test of the Leggett-Garg inequalities -- at least three distinct moments of time. Hence, the proposed inequality seems to allow one to avoid the "contextuality loophole". Violation of the proposed equality and inequality is illustrated with the behaviour of a pair of anti-correlated spins in an external magnetic field and also with the oscillations of flavour-entangled pairs of neutral pseudoscalar mesons.