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AIP conference proceedings

Star-product formalism in quantum and classical mechanics

Olga V. Man’ko, Andrei Yu. Khrennikov, Gregg S. Jaeger, M. Schlosshauer, Gregor Weihs

Published Jan 1, 2011• 0 citations
Publisher

Abstract

A review of the symplectic, optical tomography and photon number tomography is presented. Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied, and it is shown that the sets have common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vica versa, there exist tomograms of quantum states which are not admissible in classical mechanics. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively.

DOI
10.1063/1.3578698

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A review of the symplectic, optical tomography and photon number tomography is presented. Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied, and it is shown that the sets have common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vica versa, there exist tomograms of quantum states which are not admissible in classical mechanics. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively.
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A review of the tomographic-probability representation of classical and quantum states is presented. The tomographic entropies and entropic uncertainty relations are discussed in connection with ambiguities in the interpretation of the state tomograms which are considered either as a set of the probability distributions of random variables depending on extra parameters or as a single joint probability distribution of these random variables and random parameters with specific properties of the marginals. Examples of optical tomograms of photon states, symplectic tomograms, and unitary spin tomograms of qudits are given. A new universal integral inequality for generic wave function is obtained on the base of tomographic entropic uncertainty relations.
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