  We consider scattering processes where a quantum system is comprised of an
inner subsystem and of a boundary, and is subject to Haar-averaged random
unitaries acting on the boundary-environment Hilbert space only. We show that,
regardless of the initial state, a single scattering event will disentangle the
unconditional state (i.e., the scattered state when no information about the
applied unitary is available) across the inner subsystem-boundary partition.
Also, we apply Levy's lemma to constrain the trace norm fluctuations around the
unconditional state. Finally, we derive analytical formulae for the mean
scattered purity for initial globally pure states, and provide one with
numerical evidence of the reduction of fluctuations around such mean values
with increasing environmental dimension.
