Bidirectional Controlled Assisted Cloning of Arbitrary Unknown Single-Qudit States

By making use of d-dimensional Hadamard gates and d-dimensional controlled NOT gates, we first construct a five-qudit maximally entangled state, and then use it as the quantum channel to propose a new bidirectional scheme for cloning two different arbitrary unknown single-qudit simultaneously under the assistance from a state preparer and the control of a supervisor. This scheme consists of two stages: quantum teleportation and assisted cloning. To fulfil the purpose, the first stage of this scheme requires bidirectional controlled teleportation via our constructed five-qudit maximally entangled state as quantum channel, where two distant communicators can simultaneously exchange their unknown single-qudit states with unit fidelity and unit probability under the permission of the supervisor.In the second stage,the state preparer disentangles any remaining entangled states by introducing two auxiliary particles, implementing a unitary transformation and two single-particle projective measurements. Subsequently, they transmit a specific number of classical bits to two distinct communicators in order to generate two precise replicas of different unknown states. Subsequently, the above scheme is extended with a certain probability and unit fidelity by replacing the above d-dimensional quantum channel with a five-qudit non-maximally entangled state. Finally, we further analyze the expansion issues of our scheme from the perspectives of positive operator-valued measurement (POVM for short), general quantum channel and the number of state preparers and controllers, and point out that the scheme is secure.