Density Operators and Quasiprobability Distributions

The problem of expanding a density operator in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function P () of the P representation, the Wigner distribution W (), and the function 〈||〉, where |〉 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 () 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 () 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 () 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 〈||〉, 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 (, s) may be identified with the weight function P () of the P representation, with the Wigner distribution W (), and with the function 〈||〉 when the order parameter s assumes the values s=+1, 0, -1, respectively. The function W (, s) is shown to be the expectation value of the ordered operator analog of the 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 (, s) are exhibited as the order parameter s is varied continuously from s=-1, corresponding to the function 〈||〉, to s=+1, corresponding to the function P (). 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.