Demonstration of monogamy laws for Gaussian steering in optomechanics

Abstract: Secrecy and security are crucial in communication. So, quantum secret sharing protocol has recently been proposed to distribute a secret message to a set of parties, where the decoding procedure is forbidden individually, and a cooperative operation is needed. For this, quantum steering as an intriguing kind of nonlocality, is proven to be a useful resource for efficient implementations of the QSS protocol. Here, we study the distribution of Gaussian steering over an asymmetric three-mode optomechanical state. We show that a single-mode cannot be jointly steered by the two others, and further verify the monogamy inequalities of Gaussian steering. The state at hand displays genuine tripartite steering. Also, we observe one-way steering in the configuration (1vs1) mode as well as (1vs2) mode, hence, we reveal that the asymmetry introduced into the state we consider, is a necessary condition but not sufficient for reaching one way steering. As well, we detect one-way steering between two modes never directly interact. Strikingly, our model exhibits an extreme level of steering, where two single-mode cannot steer individually the third mode, while, they can collectively, which is decisive for practical execution of the QSS protocol.