For the upteenth time I have read a paper introducing the Wigner function essentially like this:

The Wigner-representation of a quantum state is a real-valued function on phase space defined

^{a }(with ) as(1)

It’s sort of like a probability distribution because the marginals reproduce the probabilities for position and momentum measurements:

(2)

and

(3)

But the reason it’s not a real probability distribution is that it can be negative.

The fact that can be negative is obviously a reason you can’t think about it as a true PDF, but the marginals property is a *terribly* weak justification for thinking about as a “quasi-PDF”. There are all sorts of functions one could write down that would have this same property but wouldn’t encode much information about actual phase space structure, e.g., the Jigner^{b } function

which tells as nothing whatsoever about how position relates to momentum.

Here is the real reason you should think the Wigner function is almost, but not quite, a phase-space PDF for a state :

- Consider an arbitrary length scale , which determines a corresponding momentum scale and a corresponding set
^{c }of coherent states . - If a measurement is performed on with the POVM of coherent states , then the probability of obtaining outcome is given by the Husimi Q function representation of :
(4)

- If can be constructed as a mixture of the coherent states , then