  A random unitary channel is one that is given by a convex combination of
unitary channels. It is shown that the conjectures on the additivity of the
minimum output entropy and the multiplicativity of the maximum output $p$-norm
can be equivalently restated in terms of random unitary channels. This is done
by constructing a random unitary approximation to a general quantum channel.
This approximation can be constructed efficiently, and so it is also applied to
the computational problem of distinguishing quantum circuits. It is shown that
the problem of distinguishing random unitary circuits is as hard as the problem
of distinguishing general mixed state circuits, which is complete for the class
of problems that have quantum interactive proof systems.
