  Birkhoff's Theorem states that doubly stochastic matrices are convex
combinations of permutation matrices. Quantum mechanically these matrices are
doubly stochastic channels, i.e. they are completely positive maps preserving
both the trace and the identity. We expect these channels to be convex
combinations of unitary channels and yet it is known that some channels cannot
be written that way. Recent work has suggested that $n$ copies of a single
channel might approximate a mixture (convex combination) of unitaries. In this
paper we show that $n(n+1)/2$ copies of a symmetric unital quantum channel may
be arbitrarily-well approximated by a mixture (convex combination) of unitarily
implemented channels. In addition, we prove that any extremal properties of a
channel are preserved over $n$ (and thus $n(n+1)/2$) copies. The result has the
potential to be completely generalized to include non-symmetric channels.
