Functional Analysis of Quantum Systems with Infinite Degree ofFir Freedom

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(1) We proved the Haag duality for half-infinite regions in pure states of 1-dim quantum spin chain.As a corollary, we see that translationally invariant pure ground states of 1-dim finite range Hamiltonians have the split property.(2)A translationally invariant pure state of a 1-dim quantum spin chain contains one copy of infinite entanglement if and only if the state does not have the split property provided that Alice and Bob systems are half-infinite regions of the 1-dim lattice.We also considered the case when the state is not pure but factor. In such cases, the state contains one copy of infinite entanglement if and only if the Bell's constant attains its maximal value for any normal states relative to the given one. In particular, the Gibbs states for short range interactions cannot contain one copy of infinite entanglement because the Bell's constant takes its minimum value 1. New examples of states which are not pure can be provided by non-equilibrium steady states(NESS).(3)We consider an analogue-of NESS for Bose systems with condensation. We found that Josephson current is non-vanishing for coupled Bosons with the same temperature.(4)We studied fluctuation property of entropy production of NESS in the sense of D.Ruelle and obtained the law of large number under the assumption that the time evolution of the system has certain mixing property(Asymptotic Abelian).We also investigated an analogue of the central limit theorem for entropy production. For some cases, the limit is not Gaussian.