  We introduce a novel measure to quantify the non-Gaussian character of a
quantum state: the quantum relative entropy between the state under examination
and a reference Gaussian state. We analyze in details the properties of our
measure and illustrate its relationships with relevant quantities in quantum
information as the Holevo bound and the conditional entropy; in particular a
necessary condition for the Gaussian character of a quantum channel is also
derived. The evolution of non-Gaussianity (nonG) is analyzedfor quantum states
undergoing conditional Gaussification towards twin-beam and de-Gaussification
driven by Kerr interaction. Our analysis allows to assess nonG as a resource
for quantum information and, in turn, to evaluate the performances of
Gaussification and de-Gaussification protocols.
