Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs

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 $p∊(1,\infty )$ 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.