Separability Of Sequential Isomorphisms On Quantum Effects

Quantum information has been one of the hottest research fields in mathematics,physics and computer science. In recent years,the researchers on operator theory and operator algebra study the unsolved problem of quantum information theory,which have become a new phenomenon in the research of quantum information theory. The Hilbert space effect algebra ε(H),that is,ε(H)={T|0≤T≤I},where,is the identity on the Hilbert space H,and is one of central concepts of quantum measurement and quantum information.In the thesis,we devote to giving the necessary and sufficient conditions of separability of sequential isomorphisms on Hilbert space effect algebras in multipartite quantum systems.We obtain the following main results.1.Let Pur(H)be the set of pure states on Hilbert space H.Let the operationbe an arbitrary sequential product on Hilbert space effect algebras ε(HA(?)HB),Φ:ε(HA(?)HB)â†' ε(HA⊕HB)is a sequential isomorphism with respect to . Then the following statements are equivalent:(â… )Φ(ε(HA)(?)ε(HB))(?)ε(HA)(?)ε(HB)ï¼›(â…¡)Φ(Pur(HA)(?)Pur(HB))(?)Pur(HA)(?)Pur(HB)ï¼›(â…¢)there exist unitary or anti-unitary operators UA:HAâ†'HA,UB:HBâ†'HB such that Φ(T)=(UA(?)UB)T(UA(?)UB)(?) for all T∈ε(HA(?)HB)ï¼›or if dim HA=dim HB,there exist unitary or anti-unitary operators UA:HBâ†'HA,UB HAâ†'HB such that Φ(T)=(UA(?)UB)θ(T)(UA(?)UB)(?) for all T∈ε(HA(?)HB),where the swap map θ is a linear map determined by θ(A(?)B)=B⊕A.2.Let ε((?)k=1nHk)be the Hilbert space effect algebra on the n-tensor product of complex Hilbert spaces (?)k=1nHk=H1(?)H2(?)...(?)Hn,n≥3.Denote by Tsa((?)k=1nHk)the space of all self-adjoint operators on (?)k=1nHk.Assume Φ:ε((?)k=1nHk)â†'ε((?)k=1nHk)is an arbitrary sequential isomorphism.Then the following statements are equivalent:(â… )Φ((?)k=1nε(Hk))(?)(?)k=1nε(Hk)ï¼›(â…¡)Φ((?)k=1nPur(Hk))(?)(?)k=1nPur(Hk)ï¼›(â…¢)there exist a permutation Ï€ of(1,2,...,n)determined by Ï€(i)=pi and linear or conjugate linear isometries Uj:Hpjâ†'Hj,j=1,2,...,n,such thatΦ(T)=(U1(?)U2(?)…(?)Un)θπ(T)(U1(?)U2(?)…(?)Un)(?)for all T∈ε((?)k=1nHk),where θπ:Tsa((?)k=1nHk)â†'Tsa((?)k=1nHk)is linear map determined by θπ(A1(?)A2(?)...(?)An)=Ap1(?)Ap2(?)...(?)An...