Maps Preserve Bregman F-divergence And Jensen F-divergence On The Set Of Finite-rank Positive Operators

Preserver problems on operator algebras have much more concerned in the study of operator theories.The study of characterizing of linear or nonlinear mappings on operator algebras that preserve the relations between operators,functions and the set of operators invariant is helpful for making the structure of operator algebras to be understand better.Bregman f-divergence mainly studies the difference between two specific elements(vectors,matrices,functions,etc.),and has a wide range of applications in quantum information science.Jensen f-divergence is also a concept with a wide range of applications.Therefore,mappings that preserve Bregman f-divergence and Jensen f-divergence has been widely concerned.Let H be an infinite separable Hilbert space and F+(H)be the set of all positive semi-definite trace class operators on H.In this paper,we mainly study the characterization of mappings that preserve Bregman f-divergence and Jensen f-divergence on the F+(H)(where f is a differentiable strictly convex function defined on the interval(0,+∞)).The properties of two kinds of divergence and the characterization of the mappings that preserve Bregman f-divergence and Jensen f-divergence are obtained on theF+(H).In Chapter 2,we characterize mappings preserving Bregman f-divergence and Umegaki relative entropy on the F+(H),respectively.For all A,B∈F+(H),the Bregman f-divergence of A and B is written as Hf(A,B)=Tr(f(A)-f(B)-f’(B)(A-B)),where f is a differentiable strictly convex function defined on the interval(0,+∞)satisfying some conditions.In this chapter,we give the characterization of mappings preserving Bregman f-divergence on the F+(H)of infinite dimensional separable Hilbert Spaces If Φ:F+(H)→(H)is a bijection satisfying Hf(Φ(A),Φ(B))=Hf(A,B)for all A,B∈F+(H),if and only if there exist either unitary operators or anti-unitary operators U on H such that Φ is of the form Φ(A)=UAU*for all A∈F+(H).Furthermore,for this special function f(that is,f is x(?)xlogx-x),the Umegaki relative entropy of A and B is written as S(A‖B)=Tr(A(logA-logB)-(A-B)).We also give the characterization of the mapping preserving Umegaki relative entropy on theF+(H).IfΦ:F+(H)→F+(H)is a bijection satisfying S(Φ(A)‖Φ(B))=S(A‖B)for all A,B∈F+(H),if and only if there exist either unitary operators or anti-unitary operators U on H such that Φ has the formΦ(A)=UAU*for all A ∈F+(H).In Chapter 3,we characterize the mapping that preserves Jensen f-divergence on the F+(H).Let λ∈(0,1),A,B∈F+(H),the Jensen f-divergence of A and B is denoted as Jf,λ(A,B)=Tr(λf(A)+(1-λ)f(B)-f(λA+(1-λ)B)),where f is a differentiable strictly convex function defined on the interval(0,+∞)satisfying some conditions.If Φ:F+(H)→F+(H)is a bijection and satisfies Jf,λ(Φ(A),Φ(B))=Jf,λ(A,B)for all A,B∈F+(H),if and only if there exist either unitary operators or anti-unitary operators U on H such that Φ has the form Φ(A)=UAU*for all A∈F+(H).