Imprecise probability for non-commuting observables

TL;DR

This paper introduces an imprecise probability framework for non-commuting quantum observables, representing joint distributions as probability intervals rather than precise values, and provides axioms for deriving these upper and lower probabilities.

Contribution

It develops a novel imprecise probability approach for non-commuting observables, extending traditional joint probability concepts in quantum mechanics.

Findings

Joint distributions are represented as probability intervals.

Axioms for upper and lower probability operators are proposed.

Framework reduces to standard probabilities for commuting observables.

Abstract

It is known that non-commuting observables in quantum mechanics do not have joint probability. This statement refers to the precise (additive) probability model. I show that the joint distribution of any non-commuting pair of variables can be quantified via upper and lower probabilities, i.e. the joint probability is described by an interval instead of a number (imprecise probability). I propose transparent axioms from which the upper and lower probability operators follow. They depend only on the non-commuting observables and revert to the usual expression for the commuting case.