Controlled and assisted cloning of two-, three- and four-qubit states with optimal quantum resources

Utilizing Hadamard gates and controlled not gates, maximally entangled states of four, five and six qubits are generated. Then, we propose respectively controlled and assisted cloning schemes of unknown two-, three- and four-qubit states via these constructed states as quantum channels. Each of the three schemes includes two stages: controlled teleportation and assisted cloning. The controlled teleportation stage of our schemes all require a single-qubit $Z$-basis measurement by controller, and need different GHZ-state measurements by senders of three schemes to complete the transmission process of their respective quantum states. In the assisted cloning stage of our schemes, after the state preparers of three schemes perform respectively appropriate GHZ-state measurements, at the senders’ locations, perfect replicas and orthogonal complementary copies of unknown states will be produced through appropriate local quantum operations. Compared with usual assisted cloning schemes, every suggested plan is extremely efficient in terms of quantum resources, as it only requires a small number of GHZ-state measurements, significantly decreasing the need for quantum operations and classical resources. Through discussion, we conclude that our schemes are safe and can be extended to assisted cloning of unknown multi-qubit GHZ-state with multiple controllers.