Positive and Negative Joint Quantum Distributions

Andreas R. Blass, Yuri G. Gurevich

This article was written for the Logic in Computer Science column in the February 2015 issue of the Bulletin of the European Association for Theoretical Computer Science. The intended audience is general computer science audience. The uncertainty principle asserts a limit to the precision with which position x and momentum p of a particle can be known simultaneously. You may know the probability distributions of x and p individually but the joint distribution makes no physical sense. Yet Wigner exhibited such a joint distribution f(x,p). There was, however, a little trouble with it: some of its values were negative. Nevertheless Wigner's discovery attracted attention and found applications. There are other joint distribution, all with negative values, which produce the correct marginal distributions of x and p. But only Wigner's distribution produces the correct marginal distributions for all linear combinations of position and momentum. We offer a simple proof of the uniqueness and discuss related issues.

Leon W. Cohen

Aleksandr V. Belinskii

A hidden-variable theory is the traditional, but not unique, basis for constructing various types of Bell's theorem. The starting point may also be a recognition of the existence of a positive-definite probability distribution function. This assumption alone is used to formulate and prove Bell's paradoxes of different types. A specific example is used to show that a formal quantum calculation can sometimes give negative values of the joint probabilities that are used in the proof. An attempt is made to identify the physical meaning of this result and an algorithm for determination of negative joint probabilities of this type is proposed.

M. D. Srinivas, E. Wolf

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.

Silviu Guiaşu