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 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*. (C) 2013 Elsevier Inc. All rights reserved.