Quantum Optics Question and Wigner Functions

[Raptor112]

I understand that Wigner function is a quasi-probability distibution as it can take negative values, but in quantum optics I see that the Q function is mentioned as often. So what is the difference between the Q function and the Wigner Function?

 

[Truecrimson]

Q-functions are always positive. This does not mean however that quantum mechanics reduces to classical statistical mechanics, because to calculate the expectation value of an observable from it, you have to integrate it against the P-function of the observable which not only can take negative values but can be "worse" than that (e.g. being more singular than a delta distribution).

Wigner, Q- and P-functions from a one-parameter family of quasi-probability distributions and you can move between them by a convolution or deconvolution of the functions with Gaussians. Generally if you represent quantum states by one of them, you have to represent observables by the quasi-probability representation dual to it. Only the Wigner representation is self-dual.

I hope this helps.