  We show that the following nontrivial necessary precondition for an
entanglement evolution equation for pure Gaussian states under one-sided
Gaussian channels holds. Suppose a Gaussian quantum channel acts on one mode of
a pure entangled multi-mode Gaussian input state. Then, for a fixed channel,
either all output states are entangled or none of them are. In other words, if
the input state is Gaussian, pure and entangled, the separability after a
one-sided Gaussian quantum channel does not depend on the input state, but only
on the channel. Furthermore, a simple linear-algebraic separability criterion
allows to decide whether a given channel destroys the entanglement of pure
entangled input states or leaves them entangled.
