  The entanglement produced by a bilinear Hamiltonian in continuous variables
has been thoroughly studied and widely used. In contrast, the physics of
entanglement resulting from nonlinear interaction described by partially
degenerate high-order Hamiltonians remains unclear. Here, we derive a hierarchy
of sufficient and necessary conditions for the positive-partial-transposition
separability of bipartite nonlinear quantum states. The proposed criteria
detect the nonpositive-partial-transposition inseparability of higher-order
moments of states, which provides a systematic framework for the
characterization of this kind of entanglement. Through numerical simulation of
cubic and quartic Hamiltonians, we demonstrate the existence and competition of
a hierarchy of entanglement witnesses, revealing the mechanism underlying such
entanglement. Our results may provide a new direction in continuous variable
quantum information processing.
