Multideviations: The hidden structure of Bell's theorems

TL;DR

This paper introduces multideviations as a new mathematical framework to identify and analyze Bell inequalities in complex quantum scenarios, leading to the discovery of new inequalities and insights into quantum violations.

Contribution

The paper develops the multideviation framework and demonstrates its effectiveness in deriving tight Bell inequalities for arbitrary scenarios, including the novel even/odd inequalities.

Findings

Bell distributions can be generated from joint distributions by deeming certain degrees of freedom unobservable.

New tight Bell inequalities, called even/odd inequalities, are derived for arbitrary event spaces.

Quantum mechanics violations of these inequalities increase with the number of systems.

Abstract

Specification of the strongest possible Bell inequalities for arbitrarily complicated physical scenarios -- any number of observers choosing between any number of observables with any number of possible outcomes -- is currently an open problem. Here I provide a new set of tools, which I refer to as "multideviations", for finding and analyzing these inequalities for the fully general case. In Part I, I introduce the multideviation framework and then use it to prove an important theorem: the Bell distributions can be generated from the set of joint distributions over all observables by deeming specific degrees of freedom unobservable. In Part II, I show how the theorem provides a new method for finding tight Bell inequalities. I then specify a set of new tight Bell inequalities for arbitrary event spaces -- the "even/odd" inequalities -- which have a straightforward interpretation when expressed in terms of multideviations. The even/odd inequalities concern degrees of freedom that are independent of those involved in parameter independence, raising the possibility of a new Bell's theorem with stronger philosophical implications. Also, contrary to expectations, the violation of the inequalities by quantum mechanics increases in size with the number of systems.