Von Neumann entropy-preserving quantum operations

For a given quantum state ρ and two quantum operations Φ and Ψ, the information encoded in the quantum state ρ is quantified by its von Neumann entropy S(ρ). By the famous Choi–Jamiołkowski isomorphism, the quantum operation Φ can be transformed into a bipartite state, the von Neumann entropy Smap(Φ) of the bipartite state describes the decoherence induced by Φ. In this Letter, we characterize not only the pairs (Φ,ρ) which satisfy S(Φ(ρ))=S(ρ), but also the pairs (Φ,Ψ) which satisfy Smap(Φ∘Ψ)=Smap(Ψ).