Von Neumann Entropy-Preserving Quantum Operations

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 $\S^{\mathrm{map}}(\Phi\circ\Psi) = \S^{\mathrm{map}}(\Psi)$.