  We prove that the closest Gaussian state to an arbitrary $N$-mode field state
through the relative entropy is built with the covariance matrix and the
average displacement of the given state. Consequently, the relative entropy of
an $N$-mode state to its associate Gaussian one is an exact distance-type
measure of non-Gaussianity. In order to illustrate this finding, we discuss the
general properties of the $N$-mode Fock-diagonal states and evaluate their
exact entropic amount of non-Gaussianity.
