  Einstein-Smoluchowski diffusion, damped harmonic oscillations, and spatial
decoherence are special cases of an elegant class of Markovian quantum Brownian
motion models that is invariant under linear symplectic transformations. Here
we prove that for each member of this class there is a preferred timescale such
that the dynamics, considered stroboscopically, can be rewritten exactly as
unitary evolution interrupted periodically by an entanglement-breaking
measurement with respect to a fixed overcomplete set of pure Gaussian states.
This is relevant to the continuing search for the best way to describe pointer
states and pure decoherence in systems with continuous variables, and gives a
concrete sense in which the decoherence can be said to arise from a complete
measurement of the system by its environment. We also extend some of the
results of Di\'{o}si and Kiefer to the symplectic covariant formalism and
compare them with the preferred timescales and Gaussian states associated with
the POVM form.
