Kirkwood-Dirac quasiprobability

The Kirkwood–Dirac quasiprobability distribution (often abbreviated KD distribution) is a complex-valued generalization of a classical joint probability distribution. It was first introduced independently by John G. Kirkwood and later by Paul Dirac as an attempt to describe quantum states in phase space using a joint distribution for noncommuting observables (like position and momentum).

It can take complex values and is therefore described as a quasiprobability distribution. Although its marginals yield valid measurement probabilities, the full distribution is generally complex and may assume negative or nonreal values. In this sense, it can be viewed as part of a generalized probabilistic theory that extends classical probability to capture intrinsically quantum features such as coherence and contextuality.