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International Journal of Theoretical Physics

Incompleteness in the Bell Theorem with an Arbitrary Number of Settings

Koji Nagata, Renata Wong, Santanu Kumar Patro, Do Ngoc Diep, Tadao Nakamura

Published Sep 29, 2020• 0 citations
Publisher
DOI
10.1007/s10773-020-04601-2

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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.