Quantum Cloning Machines of d-level System

No-cloning theorem is one of the most fundamental differences between classical and quantum information theories. It tells us that an unknown quantum state can not be copied perfectly[1]. This no-cloning theorem has important consequences for the whole quantum information processing[2]. However, no-cloning theorem does not forbid imperfect cloning, and it is interesting to know how well we can copy an unknow quantum state. Bužek and Hillery[3] introduced a universal quantum cloning machine (UQCM) for arbitrary pure input state. It produces two identical copies whose quality is independent of the input state. It was proved that the Bužek and Hillery UQCM is optimal[4, 5, 6, 7]. The general N to M optimal quantum cloning transformation was proposed by Gisin and Massar[6, 5]. And the no-cloning theorem was also extended to other cases[9, 10]. The UQCM presented by Bužek, Hillery and Gisin, Massar is for 2-level quantum system[3, 6]. The 2-dimensional Hilbert space is spanned by 2 orthonormal basis vectors |1〉, |2〉 (| ↑〉, | ↓〉). For d-level quantum system, the N to M optimal quantum cloning is formulated by Werner[7] and Keyl and Werner[8]. They have shown that the optimal cloning map to obtain M optimal clones from N identical input states is the projection of the direct product of the N input states and M − N identity states onto the symmetric subspace of M particles. The optimal fidelity is obtained from the so-called Black Cow factor. Bužek and Hillery [11] presented the universal 1 to 2 quantum cloning transformation of states in d-dimensional Hilbert space. Albeverio and Fei[12] extended this result to 1 to M cloning, and a special case for N to M cloning in which the input state is a restricted N idential d-level quantum system. In this letter, generalizing the results in Ref.[11, 12], we shall present the optimal N to M unitary cloning transformation for d-level system. The fidelity achieves the optimal fidelity given by Werner [7]. Our results recovers the previous results in Ref.[3, 6, 11, 12] for special values of N , M and d. The cloning transformation presented in this paper should be the physical implementation of the optimal cloning