Quantifying non-Gaussianity of a quantum state by the negative entropy of quadrature distributions

We propose a non-Gaussianity measure of multimode quantum state based on the negentropy quadrature distributions. Our satisfies desirable properties as measure, i.e., faithfulness, invariance under Gaussian unitary operations, and monotonicity channels. Furthermore, we find quantitative relation between our previously proposed measures defined via relative entropy Hilbert-Schmidt distance. This allows us to estimate readily by homodyne detection, which would otherwise require full quantum-state tomography.