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In this paper, we present applications of the calculus developed in Collinsand Nechita [Comm. Math. Phys. 297 (2010) 345-370] and obtain an exact formulafor the moments of random quantum channels whose input is a pure state thanksto Gaussianization methods. Our main application is an in-depth study of therandom 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] torefine the Hastings counterexample to the additivity conjecture in quantuminformation theory. This model is exotic from the point of view of randommatrix theory as its eigenvalues obey two different scalings simultaneously. Westudy its asymptotic behavior and obtain an asymptotic expansion for its vonNeumann entropy.

Gaussianization and eigenvalue statistics for random quantum channels (III)