Typicality in random quantum scattering

We consider scattering processes where a quantum system comprises an inner subsystem and 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 Lévy's lemma to constrain the trace norm fluctuations around the unconditional state. Finally, we derive analytical formulas 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.