Characterizing Bijective Maps Preserving Convex Combinations On The Closed Unit Ball Of Banach Space And Quantum Correlation Vanishing Channels

Preserver problems on operator algebras have been on characterizing maps that preserves some invariants on operator algebras for one hundred years more or less,and have been an active branch in the theory of operators and operator algebras.Recently it has been found that preservers on operator algebras can link to the topics in the quantum information theory.For example,quantum channels are trace-preserving and completely positive linear maps on density operators,and quantum gates are unitary transformations on Hilbert spaces.In the thesis,we give a characterization of bijective maps preserving segment(i.e.,preserving convex combinations in both directions)on the closed unit ball of Banach space and quantum correlation vanishing channels,we have the following results:1.X be a strictly convex Banach real-linear space with dim X>2,B1(X)the closed unit ball of X· For a bijective map ?:B1(X)?B1(X)the following statements are equivalent:(a)? is affine;(b)? preserves convex combinations in both directions,i.e.,?([x,y])=[?(x),?(y)],x,y?B1(X);(c)? has the followi'ng formx ? Ux for all x ?B1(H),where U is an invertible bounded linear isometric operator X.2.? be a quantum channel on finite dimensional systems,the following statements are equivalent:(a)there exist a family of positive operators Wi with ?Wi = I,and orthonormal basis ei,such that(b)a channel ? is a local Adiscord annihilating channel;(c)?(?)Id maps the maximal entangled state to the classical-quantum state.