Key points are not available for this paper at this time.
It is shown that, for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form. This result generalizes the one obtained by Hudson Rep. Math. Phys. 6, 249 (1974) for one-dimensional systems.
Soto et al. (Sat,) studied this question.