Optimal Quantum Cloning Machines

We present quantum cloning machines that transform $N$ identical qubits into $M>N$ identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones $M$ tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, quantum cloning machines are universal devices to translate quantum information into classical information.