  We study the high-dimensional entanglement of a photon pair transmitted
through a random medium. We show that multiple scattering in combination with
the subsequent selection of only a fraction of outgoing modes reduces the
average entanglement of an initially maximally entangled two-photon state.
Entanglement corresponding to a random pure state is obtained when the number
of modes accessible in transmission is much less than the number of modes in
the incident light. An amount of entanglement approaching that of the incident
light can be recovered by accessing a larger number of transmitted modes. In
contrast, a pair of photons in a separable state does not gain any entanglement
when transmitted through a random medium.
