Genuine Einstein-Podolsky-Rosen steering of generalized three-qubit states via unsharp measurements


 We aim to explore all possible scenarios of ($ 1 \rightarrow 2 $) (where one wing is untrusted and the other two wings are trusted) and ($ 2 \rightarrow 1 $) (where two wings are untrusted, and one wing is trusted) genuine tripartite EPR steering. The generalized Greenberger-Horne-Zeilinger (GHZ) state is shared between three spatially separated parties, Alice, Bob and Charlie. In both ($ 1 \rightarrow 2 $) and ($ 2 \rightarrow 1 $), we discuss the untrusted party and trusted party performing a sequence of unsharp measurements, respectively. For each scenario, we deduce an upper bound on the number of sequential observers who can demonstrate genuine EPR steering through the quantum violation of tripartite steering inequality. The results show that the maximum number of observers for the generalized GHZ states can be the same with that of the maximally GHZ state in a certain range of state parameters. Moreover, both the sharpness parameters range and the state parameters range in the scenario of ($ 1 \rightarrow 2 $) steering are larger than those in the scenario of ($ 2 \rightarrow 1 $) steering.