  Very recently, strongly non-Gaussian states have been observed via a direct
three-mode spontaneous parametric down-conversion in a superconducting cavity
[Phys. Rev. X 10, 011011 (2020)]. The created multi-photon non-Gaussian
correlations are attractive and useful for various quantum information tasks.
However, how to detect and classify multipartite non-Gaussian entanglement has
not yet been completely understood. Here, we present an experimentally
practical method to characterize continuous-variable multipartite non-Gaussian
entanglement, by introducing a class of nonlinear squeezing parameters
involving accessible higher-order moments of phase-space quadratures. As these
parameters can depend on arbitrary operators, we consider their analytical
optimization over a set of practical measurements, in order to detect different
classes of multipartite non-Gaussian entanglement ranging from fully separable
to fully inseparable. We demonstrate that the nonlinear squeezing parameters
act as an excellent approximation to the quantum Fisher information within
accessible third-order moments. The level of the nonlinear squeezing quantifies
the metrological advantage provided by those entangled states. Moreover, by
analyzing the above mentioned experiment, we show that our method can be
readily used to confirm fully inseparable tripartite non-Gaussian entangled
states by performing a limited number of measurements without requiring full
knowledge of the quantum state.
