Generalized Entropy-preserving Mapping On Infinite Dimensional Systems And Exchangeability Of Quantum Effect

In the past two decades,the preservation problem of the density operator on Hilbert space has been concerned by researchers.In this paper,we give the characterization of preserving generalized entropy mapping on the density operator of separable infinite dimensional Hilbert space,and obtain the necessary and sufficient conditions for its commutativity by using the sequence product of quantum effects.The main conclusions are as follows:(1)Let H be a separable infinite dimensional Hilbert space and S(H)the set of density operators,i.e.,positive and trace-one bounded linear operators on H.Assume F is the generalized entropy defined by#12 where f is a strictly convex funcion in the interval[0,1],?(?)=(?1,?2,…,?n)denotes the vector whose elements are rearranged in nonincreasing order.?:S(H)?S(H)is a surjective map preserving generalized entropy of convex combinations of density operators,i.e.F(t?+(1-t)?)=F(t?(?)+(1-t)?(?))for each ?,??S(H)and t ?[0,1],Then there exists a unitary or anti-unitary operator U on H such that?(?)=U?U*for all p ? S(H);(2)Let H be a finite dimensional Hilbert space.We shall give the relation between the sequential product of two elements and commutativity in effect algebras.