OptimalN-to-Mcloning of conjugate quantum variables

The cloning of conjugate continuous quantum variables is analyzed based on the concept of Gaussian cloning machines, i.e., transformations that yield copies that are Gaussian mixtures centered on the state to be copied. The optimality of Gaussian cloning machines that transform N identical input states into M output states is investigated, and bounds on the fidelity of the process are derived via a connection with quantum estimation theory. In particular, the optimal N-to-$M$ cloning fidelity for coherent states is found to be equal to $MN/(MN+M\ensuremath{-}N).$