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

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 focused on asymptotic approximations with two specific cases having been proven, but a general theory as yet exists. In this paper we make further progress toward this goal by employing a mix of group and category theory to show that $n(n+1)/2$ copies of an invertible unital quantum channel are isomorphic to $n$ copies of the SU(1) group. In category theory SU(1) has properties similar to those possessed by Hermitian operators in operator theory. While not a completely general result, it provides both another specific case of a proof and a new line of approach that may prove useful in the search for such a result. In addition, we prove that, in the limit of classical information, these quantum channels behave classically, as expected.