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

Faithful measure of Quantum non-Gaussianity via quantum relative entropy