A Remark on the Optimal Cloning of An N-Level Quantum System

We study quantum cloning machines (QCM) that act on an unknown N-level quantum state and make M copies. We give a formula for the maximum of the fidelity of cloning and exhibit the unitary transformations that realize this optimal fidelity. We also extend the results to treat the case of M copies from $N^\prime$ ($M>N^\prime$) identical N-level quantum systems.