Probability Measures and projections on Quantum Logics

TL;DR

This paper explores probability measures on quantum logics using a special G-map, revealing differences from classical logic in how projections are measured and compared.

Contribution

It introduces a G-map-based approach to model probability measures on quantum logics, highlighting distinctions from classical Boolean logic.

Findings

Probability measures of projections on quantum logics are not always pure.

G-maps can model logical connectives on quantum logics.

Differences between quantum and classical probability measures are characterized.

Abstract

The present paper is devoted to modelling of a probability measure of logical connectives on a quantum logic (QL), via a $G$-map, which is a special map on it. We follow the work in which the probability of logical conjunction, disjunction and symmetric difference and their negations for non-compatible propositions are studied. We study such a $ G $-map on quantum logics, which is a probability measure of a projection and show, that unlike classical (Boolean) logic, probability measure of projections on a quantum logic are not necessarilly pure projections. We compare properties of a $G$-map on QLs with properties of a probability measure related to logical connectives on a Boolean algebra.