# Are the Wigner and Husimi transforms injective?

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?