Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces.
KeywordQuantum probabilities; Dempster-Shafer probabilities; Hilbert space; Probability theory; Inequalities; Noncommutative field theory
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AbstractThe 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 H 1, H 2. 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.
CitationVourdas A (2015) Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces. Journal of Mathematical Physics, 55(8): 082107.
Link to publisher’s versionhttp://dx.doi.org/10.1063/1.4891972
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