  We address the nonGaussianity (nG) of states obtained by weakly perturbing a
Gaussian state and investigate the relationships with quantum estimation. For
classical perturbations, i.e. perturbations to eigenvalues, we found that nG of
the perturbed state may be written as the quantum Fisher information (QFI)
distance minus a term depending on the infinitesimal energy change, i.e. it
provides a lower bound to statistical distinguishability. Upon moving on
isoenergetic surfaces in a neighbourhood of a Gaussian state, nG thus coincides
with a proper distance in the Hilbert space and exactly quantifies the
statistical distinguishability of the perturbations. On the other hand, for
perturbations leaving the covariance matrix unperturbed we show that nG
provides an upper bound to the QFI. Our results show that the geometry of
nonGaussian states in the neighbourhood of a Gaussian state is definitely not
trivial and cannot be subsumed by a differential structure. Nevertheless, the
analysis of perturbations to a Gaussian state reveals that nG may be a resource
for quantum estimation. The nG of specific families of perturbed Gaussian
states is analyzed in some details with the aim of finding the maximally non
Gaussian state obtainable from a given Gaussian one.
