Bell-type inequalities for bivariate maps on orthomodular lattices

TL;DR

This paper explores Bell-type inequalities within orthomodular lattices using s-maps, revealing that simple two-proposition inequalities are always satisfied and offering new insights into multivariate maps and quantum propositions.

Contribution

It introduces the use of s-maps and j-maps for Bell-type inequalities on orthomodular lattices, providing new interpretations and showing their application to non-compatible quantum propositions.

Findings

Simple two-proposition inequalities are always satisfied.

Equivalence of various Bell-type inequalities with bivariate maps is established.

s-maps and j-maps can represent counterfactual quantum propositions.

Abstract

Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our invesigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.