When is the Wigner function of multidimensional systems nonnegative?
Francisco Soto, P. Claverie · Journal of Mathematical Physics · 1983
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.