Rapidly decaying Wigner functions are Schwartz functions

We show that if the Wigner function of a (possibly mixed) quantum state decays toward infinity faster than any polynomial in the phase space variables x and p, then so do all its derivatives, i.e., it is a Schwartz function on phase space. This is equivalent to the condition that the Husimi function is a Schwartz function, that the quantum state is a Schwartz operator in the sense of Keyl et al. [Rev. Math. Phys. 28(03), 1630001 (2016)], and, in the case of a pure state, that the wavefunction is a Schwartz function on configuration space. We discuss the interpretation of this constraint on Wigner functions and provide explicit bounds on Schwartz seminorms.