Unconditional quantum correlations do not violate Bell's inequality

TL;DR

This paper argues that quantum correlations used in Bell's inequality are conditional, and when considering complete correlations with additional randomness, they do not violate Bell's inequality, challenging traditional interpretations.

Contribution

It demonstrates that Bell's inequality violation arises from misinterpreting conditional quantum correlations as unconditional, emphasizing the importance of complete correlations in quantum measurement theory.

Findings

Conditional quantum correlations do not violate Bell's inequality.

Additional sources of randomness reduce quantum correlations.

Classical conditional correlations also do not satisfy Bell's inequality.

Abstract

In this paper I demonstrate that the quantum correlations of polarization (or spin) observables used in Bell's argument against local realism have to be interpreted as {\it conditional} quantum correlations. By taking into account additional sources of randomness in Bell's type experiments, i.e., supplementary to source randomness, I calculate (in the standard quantum formalism) the complete quantum correlations. The main message of the quantum theory of measurement (due to von Neumann) is that complete correlations can be essentially smaller than the conditional ones. Additional sources of randomness diminish correlations. One can say another way around: transition from unconditional correlations to conditional can increase them essentially. This is true for both classical and quantum probability. The final remark is that classical conditional correlations do not satisfy Bell's inequality. Thus we met the following {\it conditional probability dilemma}: either to use the conditional quantum probabilities, as was done by Bell and others, or complete quantum correlations. However, in the first case the corresponding classical conditional correlations need not satisfy Bell's inequality and in the second case the complete quantum correlations satisfy Bell's inequality. Thus in neither case we have a problem of mismatching of classical and quantum correlations. It seems that the whole structure of Bell's argument was based on unacceptable identification of conditional quantum correlations with unconditional classical correlations.