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Reports on Mathematical Physics

On some probabilistic inequalities related to the bell inequality

Enrico G. Beltrametti, Macie J. Ma̧czyński

Published Aug 1, 1993• 3 citations
Publisher
DOI
10.1016/0034-4877(93)90047-i

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Relations between Bell's inequality and noncommutativity of operators are discussed via the four operators involved in the Clauser et al inequality. The case of all operators commuting (i.e. the six commutators vanish) and the case of three out of the four operators mutually commuting (i.e. five commutators vanish) is shown to abide by the inequality. In the latter case a novel insight is unravelled. The Bell quantum bound is obeyed for the case when four commutators vanish. The probabilistic upper limit of the inequality is reviewed and shown to be 4. In any theory based on Hilbert space, the upper limit is .
In strict accordance with special relativity, locality is defined in terms of observed results. In a non-deterministic "hidden-parameter" theory, admitting "statistical independence" of the outcomes for a single pair, locality and a restriction ("primary locality") at the level of "primary probabilities", Bell's inequalities are recollected. Their violation is traced to the assumption of an algebraic structure defining relations between single-constituent "probabilistic" descriptions that cannot be interpreted as distinctive properties of the components. It is then concluded that the experimental violation of Bell's inequalities cannot be a test of non-locality, even if locality is interpreted at the unobservable level, in terms of "primary probabilities".