quasiprobability

Quasiprobability refers to a family of functions used in quantum mechanics to represent quantum states in a phase-space-like picture. Unlike classical probability distributions, quasiprobability distributions can take negative values or be more singular than ordinary functions. They are designed to reproduce correct quantum expectations and to provide insight into the behavior of noncommuting observables such as position and momentum.

Several well-known examples exist. The Wigner function is a real, normalized function on phase space whose marginals give the correct position and momentum distributions for a quantum state. The Glauber–Sudarshan P representation expresses the density operator as a weighted superposition of coherent states, but its P function can be highly singular or not a true function for nonclassical states. The Husimi Q function is a smooth, nonnegative distribution obtained from the overlap with coherent states. These and related distributions form a one-parameter family, the s-ordered (or Cahill–Glauber) quasiprobabilities W^{(s)} with s in [-1, 1], where s=−1 yields Q, s=0 yields Wigner, and s=+1 yields P.

Key features include normalization and marginals that connect to observable statistics, though not all marginals correspond to simple classical probabilities. Negativity in a quasiprobability distribution is often interpreted as a signature of nonclassicality, such as quantum interference. Quasiprobabilities have applications in quantum state tomography, quantum optics, and quantum information, offering a practical framework for visualizing and analyzing quantum states in a phase-space context.