Quasiprobability distribution

A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time-frequency analysis, and elsewhere.

Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, the ability to yield expectation values with respect to the weights of the distribution. However, they can violate the σ-additivity axiom: integrating over them does not necessarily yield probabilities of mutually exclusive states. Quasiprobability distributions also have regions of negative probability density, counterintuitively, contradicting the first axiom.

Introduction

In the most general form, the dynamics of a quantum-mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written

ρ

^

{\displaystyle {\widehat {\rho }}}

) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i.e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a manner more resembling of an ordinary function, at the expense that the function has the features of a quasiprobability distribution.