  We address the question of the multiplicativity of the maximal p-norm output
purities of bosonic Gaussian channels under Gaussian inputs. We focus on
general Gaussian channels resulting from the reduction of unitary dynamics in
larger Hilbert spaces. It is shown that the maximal output purity of tensor
products of single-mode channels under Gaussian inputs is multiplicative for
any p>1 for products of arbitrary identical channels as well as for a large
class of products of different channels. In the case of p=2 multiplicativity is
shown to be true for arbitrary products of generic channels acting on any
number of modes.
