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Mathematical Structures in Computer Science

Joint distributions, the uncertainty principle and positive distributions

Leon W. Cohen

Published Mar 28, 2014• 0 citations
Publisher

Abstract

We examine the construction of joint probabilities for non-commuting observables. We show that there are indications in standard quantum mechanics that imply the existence of conditional expectation values, which in turn implies the existence of a joint distribution. We also argue that the uncertainty principle has no bearing on the existence of joint distributions but only constrains the marginal distributions. In addition, we show that within classical probability theory there are mathematical quantities that are similar to quantum mechanical wave functions. This is shown by generalising a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function.

DOI
10.1017/s0960129512000709

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We examine the construction of joint probabilities for non-commuting observables. We show that there are indications in standard quantum mechanics that imply the existence of conditional expectation values, which in turn implies the existence of a joint distribution. We also argue that the uncertainty principle has no bearing on the existence of joint distributions but only constrains the marginal distributions. In addition, we show that within classical probability theory there are mathematical quantities that are similar to quantum mechanical wave functions. This is shown by generalising a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function.
Abstract We show that within classical probability theory there are mathematical quantities which are similar to quantum mechanical wave functions and operators. This is shown by generalizing a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function. We show that for the one dimensional case the methods of quantum mechanics for obtaining expectation values and distributions of observables follow simply. Particular difficulties arise for two non-commuting operators but nonetheless improper “quasi-distributions” can be defined and used with profit. Quasi-distributions can be thought of as two dimensional mappings of a one dimensional function. In the mathematical sense the distribution contains the same information as the wave function, since it is constructed from it and the wave function can be obtained from it uniquely. Nonetheless, an immense simplification occurs when one studies the wave function in a quasirepresentation: the physical nature of the wave function becomes much clearer. A number of explicit examples are given.
Connections between Heisenberg uncertainty relations and the existence of joint probability distributions of corresponding observables in the frame of quantum logics are investigated. It is shown that, provided the set of physical states is sufficient, uncertainly relations imply the nonexistence of joint probability distributions in any state On etudie la relation entre les relations d'incertitude de Heisenberg et l'existence de distributions de probabilites multiples des observables correspondants dans le cadre des logiques quantiques. On montre que, si l'ensemble des etats physiques est suffisant, les relations d'incertitude impliquent la non existence des distributions de probabilite multiples dans quelque etat que ce soit
The fundamental problem considered is that of the existence of a joint probability distribution for momentum and position at a given instant. The philosophical interest of this problem is that for the potential energy functions (or Hamiltonians) corresponding to many simple experimental situations, the joint “distribution” derived by the methods of Wigner and Moyal is not a genuine probability distribution at all. The implications of these results for the interpretation of the Heisenberg uncertainty principle are analyzed. The final section consists of some observations concerning the axiomatic foundations of quantum mechanics.
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.
We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.
In contrast to a widespread belief, Wigner's theorem allows the construction of true joint probabilities in phase space for distributions describing the object system as well as for distributions depending on the measurement apparatus. The fundamental role of Heisenberg's uncertainty relations in Schroedinger form (including correlations) is pointed out for these two possible interpretations of joint probability distributions. Hence, in order that a multivariate normal probability distribution in phase space may correspond to a Wigner distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schroedinger form should be satisfied.