Excluding joint probabilities from quantum theory

TL;DR

This paper investigates the challenge of defining joint probabilities for non-commuting quantum observables, showing that precise joint probabilities are impossible in two-dimensional systems, but imprecise probabilities offer consistent alternatives.

Contribution

It demonstrates the non-existence of precise joint probabilities for two non-commuting observables in qubit systems and relates this to quantum imprecise probabilities, expanding understanding beyond Bell and Kochen-Specker theorems.

Findings

No precise joint probability exists for two non-commuting observables in a two-dimensional Hilbert space.

Imprecise probabilities are consistent with quantum constraints and can incorporate measurement contexts.

Joint probabilities are not excluded if measurement contexts are considered, but they are constrained by imprecise probabilities.

Abstract

Quantum theory does not provide a unique definition for the joint probability of two non-commuting observables, which is the next important question after the Born's probability for a single observable. Instead, various definitions were suggested, e.g. via quasi-probabilities or via hidden-variable theories. After reviewing open issues of the joint probability, we relate it to quantum imprecise probabilities, which are non-contextual and are consistent with all constraints expected from a quantum probability. We study two non-commuting observables in a two-dimensional Hilbert space and show that there is no precise joint probability that applies for any quantum state and is consistent with imprecise probabilities. This contrasts to theorems by Bell and Kochen-Specker that exclude joint probabilities for more than two non-commuting observables, in Hilbert space with dimension larger than two. If measurement contexts are included into the definition, joint probabilities are not anymore excluded, but they are still constrained by imprecise probabilities.