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In the most general form, the dynamics of a quantum mechanical system are determined by a master equation - an equation of motion for the density operator (usually written ) of the system. Although it is possible to directly integrate this equation for very small systems (ie systems with few particles or degrees of freedom), this quickly becomes impossible for larger systems. For this reason, it is frequently useful to represent the density operator as a distribution over some (over-)complete operator basis. The evolution of the system is then completely determined by the evolution of a quasi-probability distribution function. This general technique has a long history, especially in the context of quantum optics. The most common examples of quasi-probability representations are the Wigner, P- and Q-functions. More recently, the positive P function and a wider class of generalized P functions have been used to solve complex problems in both quantum optics and the newer field of quantum atom optics.

Contents 1 Characteristic Functions 2 Quasi-probability Functions 3 Time Evolution and Operator Correspondences 4 Example - The Anharmonic Oscillator 5 References

Analogous to probability theory, quantum quasi-probability distributions can be written in terms of characteristic functions, from which all operator expectation values can be derived. The characteristic functions for the Wigner, Glauber P and Q distributions of an N mode system are as follows:

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Here and are vectors containing the annihilation and creation operators for each mode of the system. These characteristic functions can be used to directly evaluate expectation values of operator moments. The ordering of the annihilation and creation operators in these moments is specific to the particular characteristic function. For instance, normally ordered (annihilation operators preceding creation operators) moments can be evaluated in the following way from :

In the same way, expectation values of anti-normally ordered and symmetrically ordered combinations of annihilation and creation operators can be evaluated from the characteristic functions for the Q and Wigner distributions, respectively.

The quasi-probability functions themselves are defined as Fourier transforms of the above characteristic functions. That is,

Here and may be identified as coherent state amplitudes in the case of the Glauber P and Q distributions, but simply c-numbers for the Wigner function. Since differentiation in normal space becomes multiplication in fourier space, moments can be calculated from these functions in the following way:

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Here denotes symmetric ordering.

These relationships motivate comparisons between the distribution functions and classical probability densities. The analogy - though strong - is not perfect, as the above functions are not necessarily positive for all . Hence the term quasi-probability function.

Since each of the above transformations from through to the distribution function is linear, the equation of motion for each distribution can be obtained by performing the same transformations to . Furthermore, as any master equation which can be expressed in Lindblad form is completely described by the action of combinations of annihilation and creation operators on the density operator, it is useful to consider the effect such operations have on each of the quasi-probability functions.

For instance, consider the annihilation operator acting on . For the characteristic function of the P distribution we have

Taking the Fourier transform with respect to to find the action corresponding action on the Glauber P function, we find

By following this procedure for each of the above distributions, the following operator correspondences can be identified:

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Here s = 0, 1/2 or 1 for P, Wigner and Q distributions, respectively.

In this way, master equations can be expressed as an equations of motion of quasi-probability functions.

Consider a single-mode system evolving under the following Hamiltonian operator:

	Probability distributions [ view • talk • edit ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square)• inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • normal inverse Gaussian • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular

• H. J. Carmichael, Statistical Methods in Quantum Optics I: Master Equations and Fokker-Planck Equations, Springer-Verlag (2002).
• C. W. Gardiner, Quantum Noise, Springer-Verlag (1991).