Are the Wigner and Husimi transforms injective?

Question

I am wondering if the Wigner function is injective. By injective I mean, that, for every density matrix $\rho$, there is a different Wigner distribution. The same question applies to the Husimi distribution.

If the dynamical group is $\mathrm{SU}(2)$, the Husimi function does not recognize all states as being different, whereas the Wigner function does. How far can I "trust" the Wigner function, in general? Is there some proof of a one-to-one correspondence between density matrices and phase space distributions?

Answer 1

The answer is a resounding "yes", cf Ref. 1, provided by Groenewold in 1946, op cit, and countless emulators since.

The Husimi is completely equivalent, so, injective, to the Wigner d.f., and so the answer is ipso facto "yes" here too.

I do not understand your particular SU(2)-blindness attributed to the Husimi, but I trust it is just an artifact of some breezy specific implementation implicitly used. At least in "regular" physics of the "street", (i.e. outside the business of manufacturing instructive freak counterexamples) They are all representation changes w.r.t. each other. References:

1. Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.