Density Operators and Quasiprobability Distributions

Abstract

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 < , has bounded eigenvalues for Res = , 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.