Characterizing Multipartite non-Gaussian Entanglement for a Three-Mode Spontaneous Parametric Down-Conversion Process

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 multiphoton 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.