Cloning the quantum oscillator

A class of completely positive maps are constructed which perform an approximate cloning of arbitrary states of a multi-mode quantum oscillator. Inside this class we find the optimal maps which do the cloning with greatest accuracy. The cloning errors appear in the characteristic functions of the states as additive Gaussian noise. The construction is extended to multiple clones, and it is shown that the minimal noise has an upper bound as the multiplicity goes to infinity. It is also shown that the construction is closely related to the formalism for linear quantum amplifiers and beamsplitters used in quantum optics.