Transformations Leaving The Maximal Eigenvalue Of Quantum State Invariant

In recent years,The preservation problem of operator structure has always been one of the important fields of operator theory research.It needs different techniques to characterize when we choose different kinds of functions as invariants.In the paper.We first derive sufficient and necessary conditions for the equality of the two quantum states that use the properties of the maximum eigenvalue of the rank-one perturbed group of quantum states.Then,The characterization of the map with the maximum eigenvalue of the convex combination of infinite dimensional Hilbert space.The main conclusions in the article are as follows:Part ?For quantum states ?,??E S(H),the following statements are equivalent:(?)?=?;(?)?_max(??+(1-?)?)=Amax(??+(1-?)?)for every pure state ? and ??[0,1].Part ?Let H be a infinite dimensional Hilbert space and S(H)the set of quantum states.Denote by ?_max(A)the maximal eigenvalue of A and ?:S(H)?S(H)is a surjective map.Then the following statements are equivalent:(?)? preserves the maximal eigenvalue of convex combinations,i.e.,?_max(tp+(1-t)?)=Amax(t?(?)+(1-t)?(?))for all ?,??S(H)and t ?[0,1];(?)there exists a unitary or anti-unitary operator U on H such that ?(?)=U?U^*for all ??S(H).