  Non-Gaussian entangled states play a crucial role in harnessing quantum
advantage in continuous-variable quantum information. However, how to fully
characterize N-partite (N > 3) non-Gaussian entanglement without quantum state
tomography remains elusive, leading to a very limited understanding of the
underlying entanglement mechanism. Here, we propose several necessary and
sufficient conditions for the positive-partial-transposition separability of
multimode nonlinear quantum states resulting from high-order Hamiltonians and
successive beam splitting operations. When applied to the initial state, the
beam-splitter operations induce the emergence of different types of
entanglement mechanisms, including pairwise high-order entanglement, collective
high-order entanglement and the crossover between the two. We show numerically
that for the four-mode scenario, the threshold for the existence of
entanglement for any bipartition does not exceed the entanglement of the
original state at fixed high-order moments. These results provide a new
perspective for understanding multipartite nonlinear entanglement and will
promote their application in quantum information processing.
