Superbroadcasting of harmonic oscillators mixed states

We consider the problem of broadcasting quantum information encoded in the displacement parameter for an harmonic oscillator, from N to M>N copies of a thermal state. We show the Weyl-Heisenberg covariant broadcasting map that optimally reduces the thermal photon number, and we prove that it minimizes the noise in conjugate quadratures at the output for general input states. We find that from two input copies broadcasting is feasible, with the possibility of simultaneous purification (superbroadcasting).