Hierarchically controlling quantum teleportations

Controlling execution of certain quantum tasks in a quantum manner is necessary to ensure correctness of the final decision in case some doubts or/and wrong things might happen at the last-minute situation. As the importance of different tasks in practice may differ, controlling such tasks must be properly hierarchized: A lesser important task can be controlled in a looser way while a more important task must be put under a more severe control. Here, for concreteness, we deal with teleportations of three unknown quantum states containing information of different levels of importance. We design a scheme such that teleportation of the state of the lowest level of importance is allowed only by one agent, the general supervisor, but teleportation of the state of the middle level of importance is controlled by two agents, the general supervisor plus one more agent, and teleportation of the state of the highest level of importance requires permissions from three agents, the general supervisor plus two more agents. For that purpose, we construct quantum circuits to produce a genuine multipartite entangled state to be shared among all the authorized agents and show its relevance for the hierarchical control mentioned above.