Husimi Q representation

The Husimi Q representation, introduced by Kôdi Husimi in 1940,[1] is a quasiprobability distribution commonly used in quantum mechanics[2] to represent the phase space distribution of a quantum state such as light in the phase space formulation.

It is constructed in such a way that observables written in anti-normal order follow the optical equivalence theorem.

This means that it is essentially the density matrix put into normal order.

This makes it relatively easy to calculate compared to other quasiprobability distributions through the formula which is proportional to a trace of the operator

It produces a pictorial representation of the state ρ to illustrate several of its mathematical properties.

In fact, it can be understood as the Weierstrass transform of the Wigner quasiprobability distribution, i.e. a smoothing by a Gaussian filter, Such Gauss transforms being essentially invertible in the Fourier domain via the convolution theorem, Q provides an equivalent description of quantum mechanics in phase space to that furnished by the Wigner distribution.

Q is normalized to unity, and is non-negative definite[7] and bounded: Despite the fact that Q is non-negative definite and bounded like a standard joint probability distribution, this similarity may be misleading, because different coherent states are not orthogonal.

Two different points α do not represent disjoint physical contingencies; thus, Q(α) does not represent the probability of mutually exclusive states, as needed in the third axiom of probability theory.