Supermeasured: Violating Bell-Statistical Independence Without Violating Physical Statistical Independence

Bell's theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as fine-tuning or a conspiracy. We here point out that the common interpretation is at best physically ambiguous and at worst incorrect. The problem with the common interpretation is that Statistical Independence might be violated because of a non-trivial measure in state space, a possibility we propose to call supermeasured. We use Invariant Set Theory as an example of a supermeasured theory that violates the Statistical Independence assumption in Bell's theorem without requiring correlations between hidden variables and measurement settings (physical statistical independence).

Automated summary

Bell's theorem is often cited as evidence that quantum mechanics violates local causality and cannot be explained by hidden variables theory. However, this is only true if the hidden variables theory satisfies the assumption of Statistical Independence. The existence of correlations between measurement settings and hidden variables, which determine measurement outcomes, has been interpreted as violations of Statistical Independence. This interpretation is problematic and may be physically ambiguous or incorrect. This study proposes the concept of supermeasured theory as a possible explanation for the violation of Statistical Independence without requiring correlations between hidden variables and measurement settings.