A hierarchy of compatibility and comeasurability levels in quantum logics with unique conditional probabilities

TL;DR

This paper introduces a hierarchy of five levels of compatibility and comeasurability in quantum logics with unique conditional probabilities, clarifying their relationships and differences within quantum mechanics and related models.

Contribution

It defines a hierarchy of compatibility levels in quantum logics with unique conditional probabilities, linking them to quantum measurement processes and classical substructures.

Findings

All five levels coincide in standard quantum mechanics and von Neumann algebras.

The hierarchy clarifies distinctions between different types of measurement compatibility.

The levels range from no interference to belonging to the same Boolean algebra.

Abstract

In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lueders - von Neumann measurement process. This motivates the definition of a hierarchy of five compatibility and comeasurability levels in the abstract setting of the quantum logics with unique conditional probabilities. Their meanings are: the absence of quantum interference or influence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic (events) belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.