quasiprobabilities

Quasiprobabilities are mathematical constructions used in quantum mechanics to describe the state of a system in phase space in a way that resembles classical probability distributions. They are called quasiprobabilities because they can reproduce correct quantum statistics and marginal probabilities, yet they may take negative values or fail to be well behaved as ordinary functions. These features reflect inherently nonclassical aspects of quantum states.

The most prominent examples are the Wigner function, the Glauber-Sudarshan P function, and the Husimi Q function. The Wigner function W(x,p) provides a quasi-probability distribution over position and momentum and is constructed so that its marginals reproduce the standard quantum probability distributions for position or momentum. Unlike true probabilities, Wigner functions can assume negative values, and such negativity is commonly interpreted as a signature of nonclassicality and quantum interference.

The P function, or Glauber-Sudarshan P representation, expresses a density operator as a weighted ensemble of coherent states. For classical-like states, the P function is a true probability density; for nonclassical states it becomes highly singular or takes negative values, making it difficult to interpret as a genuine probability distribution. The Q function, or Husimi Q distribution, is obtained by smoothing the state with coherent states and is always nonnegative and smooth, but it contains less information about the state than the Wigner function.

Quasiprobabilities are widely used in quantum state tomography, quantum optics, and continuous-variable quantum information. They offer insights into nonclassicality, decoherence, and the boundary between quantum and classical behavior.