 Since the matrix ("Pi j")"i j" is a doubly stochastic matrix and log is a convex function, the above expression is

Define "r"i = ∑"j""qj Pi j".Then {"r"i} is a probability distribution.From the non-negativity of classical relative entropy, we have

The second part of the claim follows from the fact that, since -log is strictly convex, equality is achieved in

if and only if ("Pi j") is a permutation matrix, which implies "ρ" = "σ", after a suitable labeling of the eigenvectors {"vi"} and {"wi"}.