Wigner quasiprobability distribution

The Wigner quasiprobability distribution, also called the Wigner function or the Wigner-Ville distribution, is a function on phase space that stands in for a probability distribution in quantum mechanics. Eugene Wigner introduced it in 1932 to study quantum corrections to classical statistical mechanics, aiming to link the wavefunction of the Schroedinger equation to a distribution over position and momentum. It behaves like a probability density in many ways, but because of the uncertainty principle it is not a true one and can take negative values for states with no classical model, which makes negativity a useful indicator of quantum interference. It is the Wigner-Weyl transform of the density matrix, the inverse of the Weyl map from phase-space functions to operators, and smoothing it with a Gaussian gives the positive Husimi representation. Through the Wigner-Ville distribution it is also used in signal processing, time-frequency analysis, quantum optics and quantum chemistry.