Summary: 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-Jamiołkowski isomorphism, the quantum operation \(\Phi \) can be transformed into a bipartite state, the von Neumann entropy \(S^{\text{map}}(\Phi )\) 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 \(S^{\text{map}}(\Phi \circ \Psi )=S^{\text{map}}(\Psi )\).