Majorization and additivity for multimode bosonic Gaussian channels

We obtain a multimode extension of the majorization theorem for bosonic Gaussian channels, in particular, giving sufficient conditions under which the Glauber coherent states are the only minimizers for concave functionals of the output state of such a channel. We discuss direct implications of this multimode majorization for the positive solution of the famous additivity problem in the case of Gaussian channels. In particular, we prove the additivity of the output Rényi entropies of arbitrary order $p>1$. Finally, we present an alternative, more direct derivation of a majorization property of the Husimi function established by Lieb and Solovej.