Hierarchical remote preparation of an arbitrary m-qudit state with multiple agents

Based on Jin et al.’s hierarchical remote preparation scheme of an arbitrary [Formula: see text]-qudit state with three agents, we propose one scheme for multiple agents by utilizing the symmetry of cluster states. To save entanglement resource, we then put forward the other improved scheme. There exists a hierarchy among the agents in terms of their abilities which are quantified by nonconditioned fidelity. The upper-grade agent needs the assistance of all the remaining upper-grade agents and any one of the lower-grade agents to recover the target state, while the lower-grade agent requires all the other agents’ collaboration. It is important to emphasize that the recovery operation is derived by a general expression clearly revealing the relationship with the measurement results.