Minimum Rényi and Wehrl entropies at the output of bosonic channels

The minimum Rényi and Wehrl output entropies are found for bosonic channels in which the signal photons are either randomly displaced by a Gaussian distribution (classical-noise channel), or coupled to a thermal environment through lossy propagation (thermal-noise channel). It is shown that the Rényi output entropies of integer orders $z⩾2$ and the Wehrl output entropy are minimized when the channel input is a coherent state.