Density Operators and Quasiprobability Distributions

The problem of expanding a density operator $\rho$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P\left(\alpha \right)$ of the $P$ representation, the Wigner distribution $W\left(\alpha \right)$, and the function $〈\alpha \left|\rho \right|\alpha 〉$, where $|\alpha 〉$ 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 $P\left(\alpha \right)$ 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 $P\left(\alpha \right)$ 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 $W\left(\alpha \right)$ 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 $〈\alpha \left|\rho \right|\alpha 〉$, 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 $W(\alpha ,s)$ may be identified with the weight function $P\left(\alpha \right)$ of the $P$ representation, with the Wigner distribution $W\left(\alpha \right)$, and with the function $〈\alpha \left|\rho \right|\alpha 〉$ when the order parameter $s$ assumes the values $s=+1, 0, -1\mathrm{}$, respectively. The function $W(\alpha ,s)$ is shown to be the expectation value of the ordered operator analog of the $\delta$ function defined in the preceding paper. This operator is in the trace class for $\mathrm{Res}<0\mathrm{}$, has bounded eigenvalues for $\mathrm{Res}=0\mathrm{}$, and has infinite eigenvalues for $s=1\mathrm{}$. Marked changes in the properties of the quasiprobability distribution $W(\alpha ,s)$ are exhibited as the order parameter $s$ is varied continuously from $s=-1\mathrm{}$, corresponding to the function $〈\alpha \left|\rho \right|\alpha 〉$, to $s=+1\mathrm{}$, corresponding to the function $P\left(\alpha \right)$. 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.