Hierarchy of Nonlinear Entanglement Dynamics for Continuous Variables

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.

Figure 1

(a) Comparison of the criteria proposed by Nha and Zubairy and ours. ${\tilde{\nu }}_{-}^{12}$ is the minimum symplectic eigenvalue of ${\tilde{S}}^{12}$. (b) Evolution of ${\tilde{\nu }}_{-}^{12}$ (3th), ${\tilde{\nu }}_{-}^{24}$ (6th), and ${\tilde{\nu }}_{-}^{36}$ (9th) versus interaction strength $\xi$ with ${\alpha }_p=\sqrt{25}$.

Figure 2

Evolution of ${\tilde{\nu }}_{-}^{13}$ (4th), ${\tilde{\nu }}_{-}^{26}$ (8th), and ${\tilde{\nu }}_{-}^{39}$ (12th) versus $\xi$ with ${\alpha }_p=\sqrt{10}$.