  We address the quantification of nonlinearity for quantum oscillators and
introduce two measures based on the properties of the ground state rather than
on the form of the potential itself. The first measure is a fidelity-based one,
and corresponds to the renormalized Bures distance between the ground state of
the considered oscillator and the ground state of a reference harmonic
oscillator. Then, in order to avoid the introduction of this auxiliary
oscillator, we introduce a different measure based on the non-Gaussianity (nG)
of the ground state. The two measures are evaluated for a sample of significant
nonlinear potentials and their properties are discussed in some detail. We show
that the two measures are monotone functions of each other in most cases, and
this suggests that the nG-based measure is a suitable choice to capture the
anharmonic nature of a quantum oscillator, and to quantify its nonlinearity
independently on the specific features of the potential. We also provide
examples of potentials where the Bures measure cannot be defined, due to the
lack of a proper reference harmonic potential, while the nG-based measure
properly quantify their nonlinear features. Our results may have implications
in experimental applications where access to the effective potential is
limited, e.g., in quantum control, and protocols rely on information about the
ground or thermal state.
