Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces

TL;DR

This paper explores quantum probabilities within the lattice of subspaces, proposing Dempster-Shafer probabilities as a more general framework than Kolmogorov, and demonstrates their relevance through CHSH inequalities violations.

Contribution

It introduces a Dempster-Shafer probability framework for quantum systems and relates it to the lattice structure and quantum commutators, offering a novel interpretation of quantum probabilities.

Findings

Quantum probabilities violate classical additivity in the full lattice.

Dempster-Shafer probabilities do not satisfy CHSH inequalities.

Experimental violations support Dempster-Shafer interpretation.

Abstract

The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum system with d- dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities, is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2), to the subspaces H1,H2. As an application, it is shown that the proof of CHSH inequalities for a system of two spin 1/2 particles, is valid for Kolmogorov probabilities, but it is not valid for Dempster- Shafer probabilities. The violation of these inequalities in experiments, supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.