Hierarchically controlled remote preparation of an arbitrary single-qubit state by using a four-qubit $|\chi ⟩$ entangled state

In this paper, we present a scheme for Hierarchically controlled remote preparation of an arbitrary single-qubit state via a four-qubit $|\chi ⟩$ state as the quantum channel. In this scheme, a sender wishes to help three agents to remotely prepare a quantum state, respectively. The three agents are divided into two grades, that is, an agent is in the upper grade and other two agents are in the lower grade. It is shown that the agent of the upper grade only needs the assistance of any one of the other two agents for recovering the sender's original state, while an agent of the lower grade needs the collaboration of all the other two agents. In other words, the agents of two grades have different authorities to recover sender's original state.