  In this work we analyze properties of generic quantum channels in the case of
large system size. We use random matrix theory and free probability to show
that the distance between two independent random channels converges to a
constant value as the dimension of the system grows larger. As a measure of the
distance we use the diamond norm. In the case of a flat Hilbert-Schmidt
distribution on quantum channels, we obtain that the distance converges to
$\frac12 + \frac{2}{\pi}$, giving also an estimate for the maximum success
probability for distinguishing the channels. We also consider the problem of
distinguishing two random unitary rotations.
