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• W2033364080 abstract "The problem of expanding a density operator $ensuremath{rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(ensuremath{alpha})$ of the $P$ representation, the Wigner distribution $W(ensuremath{alpha})$, and the function $〈ensuremath{alpha}|ensuremath{rho}|ensuremath{alpha}〉$, where $|ensuremath{alpha}〉$ 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 $P(ensuremath{alpha})$ 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 $P(ensuremath{alpha})$ 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 ${a}^{ifmmodedaggerelsetextdaggerfi{}}$. The Wigner distribution $W(ensuremath{alpha})$ 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 $〈ensuremath{alpha}|ensuremath{rho}|ensuremath{alpha}〉$, 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 $W(ensuremath{alpha},s)$ may be identified with the weight function $P(ensuremath{alpha})$ of the $P$ representation, with the Wigner distribution $W(ensuremath{alpha})$, and with the function $〈ensuremath{alpha}|ensuremath{rho}|ensuremath{alpha}〉$ when the order parameter $s$ assumes the values $s=+1, 0, ensuremath{-}1$, respectively. The function $W(ensuremath{alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $ensuremath{delta}$ function defined in the preceding paper. This operator is in the trace class for $mathrm{Res}<0$, has bounded eigenvalues for $mathrm{Res}=0$, and has infinite eigenvalues for $s=1$. Marked changes in the properties of the quasiprobability distribution $W(ensuremath{alpha},s)$ are exhibited as the order parameter $s$ is varied continuously from $s=ensuremath{-}1$, corresponding to the function $〈ensuremath{alpha}|ensuremath{rho}|ensuremath{alpha}〉$, to $s=+1$, corresponding to the function $P(ensuremath{alpha})$. 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." @default.
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• W2033364080 title "Density Operators and Quasiprobability Distributions" @default.
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• W2033364080 doi "https://doi.org/10.1103/physrev.177.1882" @default.
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