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

The orthocomplemented modular lattice of subspaces LH (d) LH (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 LH (d) LH (d) (it is only valid within the Boolean subalgebras of LH (d) LH (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) 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) P (H1), P (H2), to the subspaces H1, H2. As an application, it is shown that the proof of the inequalities of Clauser, Horne, Shimony, and Holt 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.