  In this paper, we study the behaviour of the output of pure entangled states
after being transformed by a product of conjugate random unitary channels. This
study is motivated by the counterexamples by Hastings and Hayden-Winter to the
additivity problems. In particular, we study in depth the difference of
behaviour between random unitary channels and generic random channels. In the
case where the number of unitary operators is fixed, we compute the limiting
eigenvalues of the output states. In the case where the number of unitary
operators grows linearly with the dimension of the input space, we show that
the eigenvalue distribution converges to a limiting shape that we characterize
with free probability tools. In order to perform the required computations, we
need a systematic way of dealing with moment problems for random matrices whose
blocks are i.i.d. Haar distributed unitary operators. This is achieved by
extending the graphical Weingarten calculus introduced in Collins and Nechita
(2010).
