  We consider a quantity that is the differential relative entropy between a
generic Wigner function and a Gaussian one. We prove that said quantity is
minimized with respect to its Gaussian argument, if both Wigner functions in
the argument of the Wigner differential entropy have the same first and second
moments, i.e., if the Gaussian argument is the Gaussian associate of the other,
generic Wigner function. Therefore, we introduce the differential relative
entropy between any Wigner function and its Gaussian associate and we examine
its potential as a non-Gaussianity measure. We prove that said quantity is
faithful, invariant under Gaussian unitary operations, and find a sufficient
condition on its monotonic behavior under Gaussian channels. We provide
numerical results supporting aforesaid condition. The proposed, phase-space
based non-Gaussianity measure is complex-valued, with its imaginary part
possessing the physical meaning of the negative volume of the Wigner function.
At the same time, the real part of this measure provides an extra layer of
information, rendering the complex-valued quantity a measure of
non-Gaussianity, instead of a quantity pertaining only to the negativity of the
Wigner function. We examine the usefulness of our measure to non-Gaussian
quantum state engineering with partial measurements.
