Density Operators and Quasiprobability Distributions

Time-dependent invariants and Green functions in the probability representation of quantum mechanics

“…In the above mentioned paper it is shown that the quantum state of a system is completely determined if the classical probability distribution, w(X, µ, ν), for the variable X, is given in an ensemble of reference frames in the classical phase space. Such a function, also known as the marginal distribution function, belongs to a broad class of distributions which are determined as the Fourier transform of a characteristic function [10]. For the particular case of the variable (I.1), considered in [12]- [15], the scheme of [10] gives…”

“…In [12,13] it was recently suggested to consider quantum dynamics as a classical stochastic process described namely by a probability distribution: the so called marginal distribution function (which was discussed in a general context in [10]), associated to the position coordinate, X, taking values in an ensemble of reference frames in the phase space. Such a classical probability distribution is shown to completely describe quantum states [14,15].…”

“…Due to the Heisenberg principle, it is not possible to introduce a joint distribution function for both coordinates and momenta, while these joint probability distributions are the main tool to describe physical states in classical statistical mechanics. It is such kind of problems which has brought to the introduction of the so called quasi-probability distribution functions, such as the Wigner function [6], the Husimi function [7] and the Glauber-Sudarshan function [8], [9], later on unified into a one-parametric family [10]. Despite their wide use in quantum theory and their fundamental rôle in clarifying the link among classical and quantum aspects, these quasi-probability distributions cannot play the rôle of classical distributions since, either they allow for negative values (like the Wigner function) or they do not describe distributions of measurable variables.…”

Collapse and revival of quantum coherence for a harmonic oscillator interacting with a classical fluctuating environment

“…In this context, a thoroughly studied quantum system is the single-mode quantum harmonic oscillator interacting with a bosonic bath of oscillators. For such an open system, the decoherence time, ruling the transition from the quantum to the classical regime, may be identified by different nonclassicality criteria, which have been widely investigated [11][12][13][14][15][16][17][18][19][20][21][22][23][24] and compared [25]. Extensions to multimode systems [26][27][28][29] have been analyzed and the decoherence process has been addressed extensively [30,31].…”

Direct measurement of the quantum state of the electromagnetic field in a superconducting transmission line

“…The Wigner function contains all the information about the state of the field, and is a useful tool for studying the decoherenceinduced quantum-to-classical transition, as it provides us with a phase-space representation that can be compared to classical probability distributions [12]. For a single mode of the electromagnetic field, it is defined in terms of the respective density operatorρ as [13]:…”