CHSH Inequality on a single probability space

TL;DR

This paper derives a generalized CHSH inequality on a single probability space, demonstrating that combining data from multiple experiments does not invalidate the standard Bell inequality, thus reaffirming its interpretation in quantum mechanics.

Contribution

It provides a derivation of the standard CHSH inequality within a single probability space, countering claims that data combination invalidates Bell tests.

Findings

Standard CHSH inequality is valid when data is combined on a single probability space.

Experimental violations of Bell's inequality still challenge local realism.

The analysis supports the conventional interpretation of Bell test experiments.

Abstract

A number of papers have suggested that it is inappropriate to combine data from different experiments when undertaking experimental tests of Bell's inequalities. It has been suggested that a correct analysis, using a single probability space, leads to inequalities which are not violated by experiment. If correct, this would be contrary to the normal interpretation of such experimental data. However, in this note, a generalised Clauser-Horne-Shimony-Holt (CHSH) inequality is derived for a system of four experiments constructed on a single probability space which combines the data from the four experiments. It is shown that this leads to the standard CHSH inequality which is normally used to interpret experimental data. Thus the commonly accepted conclusion that experimental violations of Bell's inequality imply that local realistic models are inconsistent with the predictions of quantum mechanics has not been challenged.