  According to Hudson's theorem, any pure quantum state with a positive Wigner
function is necessarily a Gaussian state. Here, we make a step towards the
extension of this theorem to mixed quantum states by finding upper and lower
bounds on the degree of non-Gaussianity of states with positive Wigner
functions. The bounds are expressed in the form of parametric functions
relating the degree of non-Gaussianity of a state, its purity, and the purity
of the Gaussian state characterized by the same covariance matrix. Although our
bounds are not tight, they permit us to visualize the set of states with
positive Wigner functions.
