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 $Φ(ρ)=VρV^*$.

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