\indent I will first describe a generalization of a result by Haagerup and Thorbjornsen on the asymptotic norm of non-commutative polynomials of random matrices, in the case of unitary matrices. Then I will show how such results help us refine our understanding of the outputs of random quantum channels. In particular one obtains optimal estimates for the minimum output entropy of a large class of typical quantum channels. The first part of this talk is based on joint work with Camille Male, and the second part is based on joint work with Serban Belinschi and Ion Nechita.