Entropy-preserving maps on quantum states

In the paper, we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations. Let S(H)S(H) be the set of all quantum states (positive operators with the unit trace) on a complex Hilbert space H   with dim⁡H=n<∞dim⁡H=n<∞ and S(ρ)S(ρ) the quantum entropy of the quantum state ρ  . For arbitrary a surjective map ϕ:S(H)→S(H)ϕ:S(H)→S(H), we claim that ϕ   preserves quantum entropy of convex combinations of quantum states, i.e., satisfies S(tρ+(1−t)σ)=S(tϕ(ρ)+(1−t)ϕ(σ))S(tρ+(1−t)σ)=S(tϕ(ρ)+(1−t)ϕ(σ)) for arbitrary ρ,σ∈S(H)ρ,σ∈S(H) and t∈[0,1]t∈[0,1] if and only if there exists a unitary or anti-unitary operator U on H   such that ϕ(ρ)=UρU⁎ϕ(ρ)=UρU⁎ for all ρ∈S(H)ρ∈S(H).