  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
$\Phi$, one has $S(\Phi(\rho))=S(\rho)$ for all quantum states $\rho$ if and
only if there exists an isometric operator $V$ such that $\Phi(\rho)=V\rho
V^*$.
