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International Journal of Theoretical Physics

Problem of classical and nonclassical probabilities

Enrico G. Beltrametti, Maciej Mączyński

Published Oct 1, 1992• 7 citations
Publisher
DOI
10.1007/bf00678296

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We exploit results from the classical Stieltjes moment problem to bring out the totality of all the information regarding phase insensitive nonclassicality of a state as captured by the photon number distribution p_n. Central to our approach is the realization that n !p_n constitutes the sequence of moments of a (quasi) probability distribution, notwithstanding the fact that p_n can by itself be regarded as a probability distribution. This leads to classicality restrictions on p_n that are local in n involving p_n's for only a small number of consecutive n's, enabling a critical examination of the conjecture that oscillation in p_n is a signature of nonclassicality.
In this paper a comparison of various approaches to the problem of characterization of probabilities is presented. The notion of multidimensional probability or S-probability is introduced and it is shown that this notion can be profitably used for characterization of probability systems. The main results are given in Theorems 3 and 4. In Theorem 3 the conditions are considered under which a system of S-probabilities admits a model on a generalized event space, while in Theorem 4 the conditions under which the system admits a classical model are considered. An interpretation for S-probabilities is discussed and it is shown that Theorems 3 and 4 allow one to derive the logical structure of events from the observed probabilities.
We study nonclassicality in the product of the probabilities of noncommuting observables. We show that within the quantum theory, nonclassical states can provide larger probability product than classical states, so that nonclassical states approach the nonfluctuating states of the classical theory more closely than classical states. This is particularized to relevant complementary observables such as conjugate quadratures, phase and number, quadrature and number, and orthogonal angular momentum components.