  For a given quantum state $\rho$ and two quantum operations $\Phi$ and
$\Psi$, the information encoded in the quantum state $\rho$ is quantified by
its von Neumann entropy $\S(\rho)$. By the famous Choi-Jamio{\l}kowski
isomorphism, the quantum operation $\Phi$ can be transformed into a bipartite
state, the von Neumann entropy $\S^{\mathrm{map}}(\Phi)$ of the bipartite state
describes the decoherence induced by $\Phi$. In this Letter, we characterize
not only the pairs $(\Phi, \rho)$ which satisfy $\S(\Phi(\rho))=\S(\rho)$, but
also the pairs $(\Phi, \Psi)$ which satisfy $\S^{\mathrm{map}}(\Phi\circ\Psi) =
\S^{\mathrm{map}}(\Psi)$.
