Maps On Quantum States Preserving Quantum Entropy Of Convex Combinations

In the theory of quantum information, the topic of linear or nonlinear maps on quantum states plays an important role, for example quantum channels, NCP positive maps, maps preserving convex combinations and so on. In the paper, we give a characterization of maps on quantum states preserving quantum entropy of convex combinations. Let S(H) be the set of all quantum states on a complex Hilbert space H with dimH=n<∞and S(p) the quantum entropy of the quantum state p. For arbitrary a surjective map φ:S(H)�'S(H),(1) if dimH>2,φ preserves quantum entropy of convex combinations of quantum states, i.e., satisfies for arbitrary ρ,σ∈S(H) and t∈[0,1], S(tp+(1-t)σ)=S(tφ(ρ)+(1-t)φ(σ)) if and only if there exists a unitary or anti-unitary operator U on H such that φ(ρ)=UρU*for all ρ∈S(H);(2)if dimH=2,0preserves quantum entropy of convex combinations of quantum states if and only if there exists a unitary operator U on H such that φ(ρ)=UρU*for all ρ∈S(H) or φ(ρ)=UpTU*for all p∈S(H), where ρT is the transpose of ρ based on some orthonormal basis.