Density Operators and Quasiprobability Distributions

The problem of expanding a density operator ρ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function P(α) of the P representation, the Wigner distribution W(α), and the function <α | ρ | α>, where |α> is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function P(α) of the P representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function P(α) as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators a and a^†. The Wigner distribution W(α) is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function <α | ρ | α>, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the P representation. A parametrized integral expansion of the density operator is introduced in which the weight function W(α,s) may be identified with the weight function P(α) of the P representation, with the Wigner distribution W(α), and with the function <α | ρ | α> when the order parameter s assumes the values s=+1, 0, -1, respectively. The function W(α,s) is shown to be the expectation value of the ordered operator analog of the δ function defined in the preceding paper. This operator is in the trace class for Res<0, has bounded eigenvalues for Res=0, and has infinite eigenvalues for s=1. Marked changes in the properties of the quasiprobability distribution W(α,s) are exhibited as the order parameter s is varied continuously from s=-1, corresponding to the function <α | ρ | α>, to s=+1, corresponding to the function P(α). Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the P representation is appropriate.