  We study the limiting spectral distribution of quantum channels whose Kraus
operators are sampled as $n\times n$ random Hermitian matrices satisfying
certain assumptions. We show that when the Kraus rank goes to infinity with n,
the limiting spectral distribution (suitably rescaled) of the corresponding
quantum channel coincides with the semi-circle distribution. When the Kraus
rank is fixed, the limiting spectral distribution is no longer the semi-circle
distribution. It corresponds to an explicit law, which can also be described
using tools from free probability.
