Demonstration of monogamy laws for Gaussian steering in optomechanics

Secrecy and security are crucial in communication. Therefore, quantum secret sharing (QSS) 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, and hence, we reveal that the asymmetry introduced into the state that 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 modes cannot steer individually the third mode, while, they can collectively, which is decisive for practical execution of the QSS protocol.