On Bell-Type Inequalities in Quantum Logics

Sylvia Pulmannová

Emilio Santos

Sylvia Pulmannová, Vladimír Majerník

Three types of Bell inequalities, expressed in terms of quantum logics (orthomodular lattices) and states on them, are studied. It is shown that the two first types are equivalent to the subadditivity of the corresponding state. The third type is equivalent to the existence of a Boolean quotient B of the orthomodular lattice L and a state s̄ on B, such that s̄⋅φ=s, where s is the state on L and φ:L→B is the quotient mapping.

Krzysztof Rębilas

A fundamental Bell-type inequality is derived by referring solely to laws of logic, tertium non datur and the transitivity axiom, that are used to express the local realism assumption. The general form of the achieved inequality makes it a root of other more specific Bell-type theorems involving additional mathematical formalism and experimental conditions. Its universality is demonstrated explicitly by showing that the most famous theorems provided by Bell, Clauser, Horne, Shimony and Holt (CHSH), Clauser and Horne (CH), Eberhard, d'Espagnat and Maccone are direct corollaries of the general inequality. The method to retrieve Bell's inequalities of different kinds, as being based on a single scheme of reasoning, unifies diversity of proofs presented in the literature. Additionally, some of the known inequalities can be easily achieved in a more general form compared to their original versions and still suitable, or possibly even more convenient, for experimental investigations. By extracting local realism in the form of an explicit logical propositions and making it a cornerstone of a systematic reasoning leading to the universal Bell's inequality, the paper offers a synthetic perspective on Bell-type theorems and their logical basis.

Boris S. Tsirelson

Karl-Peter Marzlin, T. A. Osborn