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Physical Review A

Condition for a bivariate normal probability distribution in phase space to be a quantum state

J. Krüger

Published Nov 1, 1992• 7 citations
Publisher

Abstract

In order that a bivariate normal probability distribution in phase space with variances ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$ and covariance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p may correspond to a Wigner distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schr\"odinger form ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$${\mathit{p}}^{2}$\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/4 should be satisfied. The diagonalization of the corresponding density matrix entails a correspondence between the statistical and the physical properties of temperature-dependent oscillator states; the expansion of the density matrix into coherent states allows a physical interpretation in phase space.

DOI
10.1103/physreva.46.5385

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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.
Combining quantum mechanics and the normal distribution in statistics we study the coherent state from the point of view of statistics and by using the integration method within ordered product of operators. We find that the pure coherent state |z >z| exhibits a bivariate normal distribution of randon variables in (q,p) phase space, z=(q+ip)/√2, with a real k-parameter which is related to the quantization scheme, and the correlation coefficient is ik. For k=±1, |z >z| respectively is arranged as P-ordering (all P stand on the left of all Q) and Q-ordering (all Q stand on the left of all P), while in the case of k=0, |z >z| is arranged as the Weyl-ordering. In the cases of P-ordering and Q-ordering, in the classical correspondence function of |z >z||z=(q+ip)/√2 the bivariates (q,p) are correlated, only in the case of Weyl correspondece, (q,p) are independent. In other words, the Weyl ordering of operators is liable to decouple the correlation in bivariates.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
Based on the technique of integration within an ordered product of operators we investigate a completeness relation of pure states (such as the coordinate eigenstate, the momentum eigenstate and the coherent state) into normally ordered Gaussian forms. The Weyl ordering invariance under similarity transformations is employed to reveal physical meaning of a kind of normally ordered Gaussian operators, which have the similar forms to the bivariate normal distributions in statistics, i.e., the thermo mixed state density matrix.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.