The Birkhoff theorem for unitary matrices of prime-power dimension

Abstract

The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity. If the dimension n of the unitary matrix equals a power of a prime p, i.e. if n=pw, then the Birkhoff decomposition does not need all n! possible permutation matrices, as the epicirculant permutation matrices suffice. This group of permutation matrices is isomorphic to the general affine group GA(w,p) of order only pw(pw−1)(pw−p)...(pw−pw−1)≪(pw)!.