On the role of joint probability distributions of incompatible observables in Bell and Kochen-Specker Theorems

TL;DR

This paper investigates whether Bell and Kochen-Specker theorems hold when assuming local realism without requiring joint probability distributions for incompatible observables, concluding that standard inequalities still prevent loopholes.

Contribution

It introduces a realist model avoiding joint distributions for incompatible observables and demonstrates that this does not undermine the theorems' conclusions.

Findings

Derived a CHSH inequality valid for finite ensembles.

Showed that divergent sequences with convergent marginals do not create loopholes.

Analyzed Hardy's paradox under noncontextual realism constraints.

Abstract

We analyze the validity of Bell and Kochen-Specker theorems under local (or noncontextual) realism but avoiding an assumption of the existence of a joint probability distribution for incompatible observables. We formulate a realist model which complies with this requirement. This is obtained by employing divergent sequences that nevertheless have marginals which are convergent. We find that under standard reasonable assumptions this possibility does not lead to a loophole of those theorems, by deriving a short of CHSH inequality valid for any finite size ensemble. Moreover, we analyze a Hardy's paradox setting where noncontextual realism imposes the existence of joint probabilities for incompatible observables.