The Birkhoff theorem for unitary matrices of arbitrary dimensions
  It was shown recently that Birkhoff's theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension. 
Stijn De Baerdemacker, Alexis De Vos, Lin Chen, Li Yu
math-ph math.MP quant-ph