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Birkhoff's Theorem states that doubly stochastic matrices are convexcombinations of permutation matrices. Quantum mechanically these matrices aredoubly stochastic channels, i.e. they are completely positive maps preservingboth the trace and the identity. We expect these channels to be convexcombinations of unitary channels and yet it is known that some channels cannotbe written that way. Recent work has suggested that $n$ copies of a singlechannel might approximate a mixture (convex combination) of unitaries. In thispaper we show that $n(n+1)/2$ copies of a symmetric unital quantum channel maybe arbitrarily-well approximated by a mixture (convex combination) of unitarilyimplemented channels. In addition, we prove that any extremal properties of achannel are preserved over $n$ (and thus $n(n+1)/2$) copies. The result has thepotential to be completely generalized to include non-symmetric channels.

A category theoretic approach to asymptotic quantum channel approximation and Birkhoff's Theorem