Detection of quantum steering in multipartite continuous-variable Greenberger-Horne-Zeilinger-like states

The multipartite entangled state has drawn broad attention for both foundations of quantum mechanics and applications in quantum information processing. Here, we study the spatially separated N-partite continuous-variable Greenberger-Horne-Zeilinger-like states, which can be produced by a linear optical network with squeezed light and N - 1 beamsplitters. We investigate the properties of multipartite Einstein-Podolsky-Rosen steering possessed by those states, and find that the steering of a given quantum mode is allowed when not less than half of the modes within the states take part in the steering group. This is certified by the detection of the correlation between position and momentum quadratures of the steered mode and a combination of quadratures of other modes inside the steering group. The steering is evidenced by the high correlation where the steering group can infer the quadratures of the steered mode to high precision, i.e., below the quantum limit for the position and momentum quadratures of the steered quantum mode. We also examine the influence of inefficiency on the multipartite steering, and derive the threshold of the loss tolerance. Furthermore, we discuss the collective N-partite steering induced by the asymmetric loss on beams, which exists when a given quantum mode can only be steered by all the remaining N - 1 modes collaboratively. The present multipartite steering correlation may have potential applications in certain quantum information tasks where the issue of trust is important, such as one-sided device-independent quantum secret sharing.