Von Neumann entropy and majorization

We consider the properties of the Shannon entropy for two probability distributions which stand in the relationship of majorization. Then we give a generalization of a theorem due to Uhlmann, extending it to infinite dimensional Hilbert spaces. Finally we show that for any quantum channel Φ, one has S(Φ(ρ))=S(ρ) for all quantum states ρ if and only if there exists an isometric operator V such that Φ(ρ)=VρV∗.