Directional steering as a sufficient and necessary condition for Gaussian entanglement swapping: Application to distant optomechanical oscillators

Abstract

Einstein-Podolsky-Rosen (EPR) steering is a quantum nonlocal effect that characterizes the ability to remotely control quantum states of one system via local measurements on another distant system entangled with the controlled one. Entanglement swapping is an effective way to realize distant entanglement, which is useful for building quantum networks. Previous studies show that the presence of entanglement of subsystems in mixed states is merely necessary for achieving entanglement swapping. In this paper, we find that for Gaussian entanglement swapping with two pairs of entangled modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}\phantom{\rule{4pt}{0ex}}(j=1,2)$, the existence of directional EPR steering from the modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}$ under homodyne detection to the other modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ is a sufficient and necessary condition for achieving swapped entanglement between modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{2}$, and moreover this achievement is independent of the reverse steering from modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}$. We further reveal that this is because the steering in that direction enables the amplitude and phase squeezing of the two composite modes of modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{2}$ via homodyne detection, which is necessary and sufficient for swapped entanglement. As a concrete example, we next investigate light-mechanical steering in a dispersively or dissipatively coupled optomechanical system, and we consider the generation of entanglement between two distant optomechanical oscillators via entanglement swapping by utilizing steering. It is interesting to find that even without the constraint that the cavity linewidth should be smaller than the mechanical frequency, robust mechanical entanglement can be obtained in the steady-state regime.