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2007 Fiscal Year Final Research Report Summary

Functional Analysis of Quantum Systems with Infinite Degree ofFir Freedom

Research Project

Project/Area Number 16340050
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKyushu University

Principal Investigator

MATSUI Taku  Kyushu University, Faculty of Mathematics, Professor (50199733)

Co-Investigator(Kenkyū-buntansha) TASAKI Shuichi  Waseda University, Faculty of Science and Engineering, Professor (10260150)
HIROKAWA Masako  Okayama University, Department of Mathematics, Professor (70282788)
NAKAYASHIKI Atsushi  Kyushu University, Faculty of Mathematics, Associate Professor (10237456)
UEDA Yoshimichi  Kyushu University, Faculty of Mathematics, Associate Professor (00314724)
HIROSHIMA Fumio  Kyushu University, Faculty of Mathematics, Associate Professor (00330358)
Project Period (FY) 2004 – 2007
KeywordsQuantum Spin Chain / Haag duality / Entanglement / BEC / Snlit Proy / NESS / Entropy Production
Research Abstract

(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.

  • Research Products

    (15 results)

All 2008 2007 2006 2005 2004 Other

All Journal Article (10 results) (of which Peer Reviewed: 3 results) Presentation (5 results)

  • [Journal Article] On Haag Duality for Pure States of Quantum Spin Chains2008

    • Author(s)
      M.Keyl, et. al.
    • Journal Title

      Rev.Math.Phys

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Entanglement Haag-duality and type properties of infinite quantum spin chains2007

    • Author(s)
      M.Keyl, et. al.
    • Journal Title

      Rev.Math.Phys. 18

      Pages: 935-970

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] KMS states and branched points2007

    • Author(s)
      Izumi, Masaki, Tsuyoshi, Kajiwara, Watatani, Yasuo
    • Journal Title

      Ergodic Theory Dynam. Systems 27, 26

      Pages: 1887-1918

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Regularities of ground states of quantum field models2007

    • Author(s)
      Arai, Asao, Hirokawa, Masao, Hiroshima, Fumio
    • Journal Title

      J., Kyushu Math 61

      Pages: 443-456

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] BEC of Free Bosons on Networks2006

    • Author(s)
      T.Matsui
    • Journal Title

      Infinite dim.analysis and Quant.Prob. 9

      Pages: 1-26

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Entanglement Haag-duality and type properties of infinite quantum spin chains2006

    • Author(s)
      M., Keyl, T., Matsui, D., Schlingemann, R. F., Werner
    • Journal Title

      Rev. Math. Phys.18 9

      Pages: 1-26

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On entropy production of a one-dimensional lattice conductor2006

    • Author(s)
      Shuichi, Tasaki
    • Journal Title

      Quantum information V 1

      Pages: 203-217

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Nonequilibrium steady states with Bose-Einstein condensates2005

    • Author(s)
      S., Tasaki, T., Matsui
    • Journal Title

      Stochastic analysis : classical and quantum

      Pages: 211-227

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On Haag Duality for Pure States of Quantum Spin Chains

    • Author(s)
      Kely, M., et. al.
    • Journal Title

      Rev. Math. Phys. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Fiber Hamiltonians in non-relativistic quantum electrodynamics

    • Author(s)
      Hiroshima, Fumio
    • Journal Title

      J. Funct. Anal 252

      Pages: 314-355

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Entanglement in infinite quantum systems2007

    • Author(s)
      T., Matsui
    • Organizer
      Mathematical Analysis of Quantum Systems
    • Place of Presentation
      Dublin Institute for Advanced Studies
    • Year and Date
      2007-04-05
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] BEC on Networks2006

    • Author(s)
      T.Matsui
    • Organizer
      QUANTUM PROBABILITY:NONCOMMUTATIVE ANALYSIS AND APPLICATIONS
    • Place of Presentation
      ノチンガム大学
    • Year and Date
      2006-07-20
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] BEC on Networks2006

    • Author(s)
      T., Matsui
    • Organizer
      QUANTUM PROBABILITY : NONCOMMUTATIVE
    • Place of Presentation
      University of Nottingham, UK
    • Year and Date
      2006-07-20
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] BEC and NESS on Lattices2005

    • Author(s)
      T., Matsui
    • Organizer
      Open Quantum Systems
    • Place of Presentation
      Schroedinger Institute, Wien, Austria
    • Year and Date
      2005-03-15
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] On the absence of non-periodic ground states for the antiferromagnetic XXZ model2004

    • Author(s)
      T., Matsui
    • Organizer
      Workshop on dynamics in statistical Mechanics
    • Place of Presentation
      Centre de Recherche Mathematique, Montreal, Canada
    • Year and Date
      2004-08-03
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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