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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 $§(ρ)$. By the famous Choi-Jamiołkowski isomorphism, the quantum operation $Φ$ can be transformed into a bipartite state, the von Neumann entropy $§^{\mathrm{map}}(Φ)$ of the bipartite state describes the decoherence induced by $Φ$. In this Letter, we characterize not only the pairs $(Φ, ρ)$ which satisfy $§(Φ(ρ))=§(ρ)$, but also the pairs $(Φ, Ψ)$ which satisfy $§^{\mathrm{map}}(Φ\circΨ) = §^{\mathrm{map}}(Ψ)$.