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International Journal of Theoretical Physics

Operational Phase-Space Probability Distribution in Quantum Communication Theory

Masashi Ban

Published Jun 1, 2002• 0 citations
Publisher
DOI
10.1023/a:1016039311570

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It is shown that phase-space distribution functions that characterize a single-particle quantum system obey a certain "generalized non-negativity condition," which reflects the fact that the density operator is a positive operator. A corresponding criterion is obtained for the associated characteristic functions and is found to resemble, to some extent, Bochner's theorem of classical probability theory. Necessary and sufficient conditions on a phase-space representation of quantum mechanics are also derived, which ensure that all the possible distribution functions in that representation are non-negative; but it is also shown that such distribution functions are not joint probabilities for position and momentum. In fact, our results readily provide a new proof of theorems of Wigner, and of Cohen and Margenau, which imply that quantum mechanics cannot be formulated as a stochastic theory in phase space.
Recently, quantum models of carrier transport in semiconductors have been proposed on the basis of the Wigner formulation of quantum mechanics. Such models are demanded by the semiconductor industry due to the fast transition from microelectronics to nanoelectronics. However, the Wigner function cannot be interpreted as a quantum analog of a distribution function, since it is not a probability density. In this work we show how this difficulty could be overcome by introducing a non standard quantum theory, based on the concept of Positive-Operator-Valued (POV) measures [3–5,7].Such a theory is demanded by the semiconductor industry due to the fast transition from microelectronics to nanoelectronics. The Wigner function cannot be strictly interpreted as a quantum analog of a distribution function, because it is not a probability density. In this communication we show that this difficulty can be overcome by introducing a non standard quantum theory, based on the concept of Positive-Operator-Valued (POV) measures rather than on Projector-Valued (PV) measures. The introduction of this more general class of mathematical objects accounts for the stochastic effect of measure errors in the effective evaluation of observables described by PV measures [1, 2]. Following Prugovecki [3], we show that, on an appropriately defined stochastic phase space, it is possible to introduce a “quantum distribution function” whose marginals give the desired macroscopic quantities, and which does not present the disadvantages of the Wigner function. We present the resulting quantum Boltzmann equations for an ensemble of free carriers, and discuss extensions to the case of carriers in a semiconductor.