  We characterize the class of all physical operations that transform Gaussian
states to Gaussian states. We show that this class coincides with that of all
operations which can be performed on Gaussian states using linear optical
elements and homodyne measurements. For bipartite systems we characterize the
processes which can be implemented by local operations and classical
communication, as well as those that can be implemented using positive partial
transpose preserving maps. As an application, we show that Gaussian states
cannot be distilled by local Gaussian operations and classical communication.
We also define and characterize positive (but not completely positive) Gaussian
maps.
