  We introduce a measure of quantum non-Gaussianity (QNG) for those quantum
states not accessible by a mixture of Gaussian states in terms of quantum
relative entropy. Specifically, we employ a convex-roof extension using all
possible mixed-state decompositions beyond the usual pure-state decompositions.
We prove that this approach brings a QNG measure fulfilling the properties
desired as a proper monotone under Gaussian channels and conditional Gaussian
operations. As an illustration, we explicitly calculate QNG for the noisy
single-photon states and demonstrate that QNG coincides with non-Gaussianity of
the state itself when the single-photon fraction is sufficiently large.
