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 ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. 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
Res<0, has bounded eigenvalues for
Res=0, and has infinite eigenvalues for
s=1. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ 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.
The phase-space quasiprobability distribution formalism for representing quantum states provides practical tools for various applications in quantum optics such as identifying the nonclassicality of quantum states. We study filter functions that are introduced to regularize the Glauber-Sudarshan
P function. We show that the quantum map associated with a filter function is completely positive and trace preserving and hence physically realizable if and only if the Fourier transform of this function is a probability density distribution. We also derive a lower bound on the fidelity between the input and output states of a physical quantum filtering map. Therefore, based on these results, we show that any quantum state can be approximated, to arbitrary accuracy, by a quantum state with a regular Glauber-Sudarshan
P function. We propose applications of our results for estimating the output state of an unknown quantum process and estimating the outcome probabilities of quantum measurements.
Phase-space distributions are widely applied in quantum optics to access the nonclassical features of radiations fields. In particular, the inability to interpret the Glauber-Sudarshan distribution in terms of a classical probability density is the fundamental benchmark for quantum light. However, this phase-space distribution cannot be directly reconstructed for arbitrary states, because of its singular behavior. In this work, we perform a characterization of the Glauber-Sudarshan representation in terms of distribution theory. We address important features of such distributions: (i) the maximal degree of their singularities is studied, (ii) the ambiguity of representation is shown, and (iii) their dual space for nonclassicality tests is specified. In this view, we reconsider the methods for regularizing the Glauber-Sudarshan distribution for verifying its nonclassicality. This treatment is supported with comprehensive examples and counterexamples.
Marginal properties of arbitrary s-ordered quasiprobability distribution fucntions are investigated. The positivity of marginal distributions for all quasiprobabilities, interpolating between the Wigner function and the Q representation of the density operator (s\ensuremath{\le}0), is proven. We also show that, on the contrary, marginal distributions of the remaining quasiprobabilities, interpolating between the Wigner function and the P representation (s>0), can take on negative values. General formulas for the marginal distributions are given, and their relations to the actual quantum-mechanical probability distributions for position and momentum are established. Our results provide an interesting generalization of some recent results concerning marginal properties of the P representation [G. S. Agarwal, Opt. Commun. 95, 109 (1993)].
In this chapter, we will develop methods for mapping operator equations to equivalent c-number equations. This results in a continuous phase-space representation of a many-body quantum system, using phase-space distributions instead of density matrices. In order to do this, we will first find c-number representations of operators. We have already seen one such representation, the Glauber–Sudarshan P -representation for the density matrix. We now introduce several more such representations. The main focus will be on the truncated Wigner representation, valid at large photon number, and the positive P -representation, which uses a double-dimensional phase space and exists as a positive probability for all quantum density matrices. Phase-space techniques have a great advantage over conventional matrix-type solutions to the Schrödinger equation, in that they do not have an exponential growth in complexity with mode and particle number. Instead, the equations that describe the dynamics of these c-number representations of the density matrix are Fokker–Planck equations. These have equivalent stochastic differential equations, which behave as c-number analogs of Heisenberg-picture equations of motion. This means that problems that would be essentially impossible to solve using conventional number-state representations can be transformed into readily soluble differential equations. In many cases, no additional approximations, such as perturbation theory or factorization assumptions, are needed.