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 ) = V ρ V ∗ .