The Bell inequalities: identifying what is testable and what is not

TL;DR

This paper re-derives Bell inequalities in general forms, showing they are purely mathematical and independent of experimental test, and clarifies what aspects of correlations are testable in quantum systems.

Contribution

It demonstrates that Bell inequalities are mathematical results independent of experimental conditions and clarifies the distinction between testable correlations and their mathematical forms.

Findings

Bell inequalities are independent of experimental setups.

Correlations vary with physical systems and configurations.

Quantum probabilities do not violate the derived inequalities.

Abstract

The Bell inequalities in three and four correlations are re-derived in general forms showing that three and four data sets, respectively, identically satisfy them regardless of whether they are random, deterministic, measured, predicted, or some combination of these. The Bell inequality applicable to data is thus a purely mathematical result independent of experimental test. Correlations of simultaneously cross-correlated variable pairs do not in general all have the same form, and vary with the physical system considered and its experimental configuration. It is the form of correlations of associated data sets that may be tested, and not whether they satisfy the Bell inequality. In the case of pairs of spins or photons, a third measured or predicted value requires a different experimental setup or predictive computation than is used to obtain data from pairs alone. This is due to the quantum non-commutation of spin and photon measurements when there is more than one per particle of a pair. The Wigner inequality for probabilities, with different probabilities for different variable pairs, may be obtained from the four variable Bell inequality under a simple symmetry condition. Neither the probability or correlation inequality is violated by correlations computed from quantum probabilities based on non-commutation.