We investigate the equivalence of a reversible channel and a channel that is an isometry. Moreover, we study several equivalent characterizations of reversible channels, from views of adjoint channels, complementary channels, quantum fidelity and Choi representations of channels, respectively. Also, a channel that is an isometry has extension properties from the Banach space of all trace-class operators to all Schatten p-class operators. Finally, we get that the quantum relative entropy between any two states can never change by applying a channel that is an isometry to both states.
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