  In this paper, we present applications of the calculus developed in Collins
and Nechita [Comm. Math. Phys. 297 (2010) 345-370] and obtain an exact formula
for the moments of random quantum channels whose input is a pure state thanks
to Gaussianization methods. Our main application is an in-depth study of the
random matrix model introduced by Hayden and Winter [Comm. Math. Phys. 284
(2008) 263-280] and used recently by Brandao and Horodecki [Open Syst. Inf.
Dyn. 17 (2010) 31-52] and Fukuda and King [J. Math. Phys. 51 (2010) 042201] to
refine the Hastings counterexample to the additivity conjecture in quantum
information theory. This model is exotic from the point of view of random
matrix theory as its eigenvalues obey two different scalings simultaneously. We
study its asymptotic behavior and obtain an asymptotic expansion for its von
Neumann entropy.
