Quantum probabilities as DempsterShafer probabilities in the lattice of subspaces.
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201408Author
Vourdas, ApostolosRights
© 2014 AIP Publishing. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. The following article appeared in the Journal of Mathematical Physics and may be found at http://dx.doi.org/10.1063/1.4891972.PeerReviewed
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The orthocomplemented modular lattice of subspaces L[H(d)] , of a quantum system with ddimensional 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 DempsterShafer 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 DempsterShafer probabilities. The violation of these inequalities in experiments supports the interpretation of quantum probabilities as DempsterShafer probabilities.Version
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Vourdas A (2015) Quantum probabilities as DempsterShafer probabilities in the lattice of subspaces. Journal of Mathematical Physics, 55(8): 082107.Link to publisher’s version
http://dx.doi.org/10.1063/1.4891972Type
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