Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces
The orthocomplemented modular lattice of subspaces ${\cal L}[H(d)]$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 ${\cal L}[H(d)]$L[...
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Veröffentlicht in: | Journal of mathematical physics 2014-08, Vol.55 (8), p.1 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | The orthocomplemented modular lattice of subspaces ${\cal L}[H(d)]$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 ${\cal L}[H(d)]$L[H(d)] (it is only valid within the Boolean subalgebras of ${\cal L}[H(d)]$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 ${\mathfrak {D}}(H_1, H_2)$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 ${\mathfrak {P}}(H_1), {\mathfrak {P}}(H_2)$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. |
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ISSN: | 0022-2488 1089-7658 |
DOI: | 10.1063/1.4891972 |