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Matrix/operator inequalities and applications to quantum information and free probability

Research Project

Project/Area Number 26400103
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionTohoku University

Principal Investigator

Hiai Fumio  東北大学, 情報科学研究科, 名誉教授 (30092571)

Research Collaborator UEDA YOSHIMICHI  九州大学, 大学院数理学研究院, 准教授
Bourin Jean-Christophe  Universite de Franche Comte
Ruskai M. B.  University of Massachusetts Lowell, Emeritus Professor
Audenaert K. M. R.  Royal Holloway, University of London
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords関数解析学 / 作用素 / 行列解析 / 作用素環 / 量子情報 / 自由確率論 / 作用素平均
Outline of Final Research Achievements

In the course of this research project we obtained various results in the three subjects of matrix/operator analysis, quantum information, and free probability. In matrix/operator analysis, we studied the higher order extension, called k-tone functions, of operator momotone and concave functions in connection with the kth derivative of functional calculus. We introduced the notion of symmetric anti-norms for matrices/operators and discuseed the Jensen-Minkowski type cancavity and superadditivity inequalities of matrix trace/norm functions involving operator means and symmetric anti-norms. We also studied extensions of Araki's and Ando-Hiai's log-majorizations from different aspects.

In quantum information, we studied contraction coefficients for quantum channels with respect to quantum f-divergences and quantum monotone metrics. In free probability, we studied the orbital free entropy and free Fisher information from the viewpoint of variational principle based on Legendre transform.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (30 results)

All 2017 2016 2015 2014 Other

All Int'l Joint Research (6 results) Journal Article (13 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 13 results,  Acknowledgement Compliant: 10 results) Presentation (11 results) (of which Int'l Joint Research: 1 results,  Invited: 11 results)

  • [Int'l Joint Research] Royal Holloway ロンドン大学(英国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] ブダペスト工業経済大学(ハンガリー)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] 上海大学(中国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Sungkyunkwan 大学(韓国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] ミュンヘン工科大学(ドイツ)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research]

    • Related Report
      2016 Annual Research Report
  • [Journal Article] On an eigenvalue inequality involving the Hadamard product2017

    • Author(s)
      F. Hiai and M. Lin
    • Journal Title

      Linear Algebra Appl.

      Volume: 515 Pages: 313-320

    • DOI

      10.1016/j.laa.2016.11.017

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Generalized log-majorization and multivariate trace inequalities2017

    • Author(s)
      F. Hiai, R. Konig and M. Tomamichel
    • Journal Title

      Ann. Henri Poincare

      Volume: 印刷中 Issue: 7 Pages: 2499-2521

    • DOI

      10.1007/s00023-017-0569-y

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Log-majorization and Lie-Trotter formula for the Cartan barycenter on probability measure spaces2017

    • Author(s)
      F. Hiai and Y. Lim
    • Journal Title

      J. Math. Anal. Appl.

      Volume: 印刷中 Issue: 1 Pages: 195-211

    • DOI

      10.1016/j.jmaa.2017.03.027

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Reciprocal Lie-Trotter formula2016

    • Author(s)
      K. M. R. Audenaert and F. Hiai
    • Journal Title

      Linear and Multilinear Algebra

      Volume: 64 Issue: 6 Pages: 1220-1235

    • DOI

      10.1080/03081087.2015.1082957

    • Related Report
      2016 Annual Research Report 2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Contraction coefficients for noisy quantum channels2016

    • Author(s)
      F. Hiai and M. B. Ruskai
    • Journal Title

      J. Math. Phys.

      Volume: 57 Issue: 1

    • DOI

      10.1063/1.4936215

    • Related Report
      2016 Annual Research Report 2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Concavity of certain matrix trace and norm functions. II2016

    • Author(s)
      F. Hiai
    • Journal Title

      Linear Algebra Appl.

      Volume: 496 Pages: 193-220

    • DOI

      10.1016/j.laa.2015.12.032

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A generalization of Araki's log-majorization2016

    • Author(s)
      F. Hiai
    • Journal Title

      Linear Algebra Appl.

      Volume: 501 Pages: 1-16

    • DOI

      10.1016/j.laa.2016.03.023

    • Related Report
      2016 Annual Research Report 2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Orbital free pressure and its Legendre transform2015

    • Author(s)
      F. Hiai and Y. Ueda
    • Journal Title

      Comm. Math. Phys.

      Volume: 334 Issue: 1 Pages: 275-300

    • DOI

      10.1007/s00220-014-2135-5

    • NAID

      120006714548

    • Related Report
      2015 Research-status Report 2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Anti-norms on finite von Neumann algebras2015

    • Author(s)
      J.-C. Bourin and F. Hiai
    • Journal Title

      Publ. Res. Inst. Math. Sci.

      Volume: 51 Issue: 2 Pages: 207-235

    • DOI

      10.4171/prims/153

    • Related Report
      2015 Research-status Report 2014 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Operator k-tone functions and analytic functional calculus2015

    • Author(s)
      F. Hiai
    • Journal Title

      Linear and Nonlinear Analysis

      Volume: 1

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Higher order extension of Loewner's theory: Operator k-tone functions2014

    • Author(s)
      U. Franz, F. Hiai and E. Ricard
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: 366

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Conic structure of the non-negative operator convex functions on (0,∞)2014

    • Author(s)
      U. Franz and F. Hiai
    • Journal Title

      Ann. Funct. Anal.

      Volume: 5

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Jensen and Minkowski inequalities for operator means and anti-norms2014

    • Author(s)
      J.-C. Bourin and F. Hiai
    • Journal Title

      Linear Algebra Appl.

      Volume: 456 Pages: 22-53

    • DOI

      10.1016/j.laa.2014.05.030

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] Generalized log-majorization and multivariate Golden-Thompson type inequalities2016

    • Author(s)
      F. Hiai
    • Organizer
      2016 China-Korea International Conference on Matrix Theory with Applications
    • Place of Presentation
      上海(中国)
    • Year and Date
      2016-12-30
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Different quantum f-divergences and the reversibility of quantum operations2016

    • Author(s)
      F. Hiai
    • Organizer
      Workshop on Quantum Information Theory and Mathematical Physics
    • Place of Presentation
      ブダペスト(ハンガリー)
    • Year and Date
      2016-09-19
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Reversibility of stochastic maps via quantum divergences2016

    • Author(s)
      F. Hiai
    • Organizer
      17th Workshop on Non-commutative Harmonic Analysis
    • Place of Presentation
      ベドレボ(ポーランド)
    • Year and Date
      2016-07-29
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] A generalization of Araki's log-majorization2016

    • Author(s)
      F. Hiai
    • Organizer
      The 20th Conference of the International Linear Algebra Society
    • Place of Presentation
      ルーヴェン(ベルギー)
    • Year and Date
      2016-07-15
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quantum divergences and reversibility2016

    • Author(s)
      F. Hiai
    • Organizer
      Workshop "Mathematical Aspects in Current Quantum Information Theory"
    • Place of Presentation
      テジョン(韓国)
    • Year and Date
      2016-02-15
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Concavity/convexity of certain matrix trace and norm functions2015

    • Author(s)
      F. Hiai
    • Organizer
      Workshop on Quantum Information Theory and Related Topics
    • Place of Presentation
      ハノイ(ベトナム)
    • Year and Date
      2015-09-01
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Anti Lie-Trotter formula2015

    • Author(s)
      F. Hiai
    • Organizer
      International Workshop: Algebraic and Analytic Aspects of Quantum Levy Processes
    • Place of Presentation
      グライフスワルド(ドイツ)
    • Year and Date
      2015-03-12
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Powers of nonnegative matrices2014

    • Author(s)
      F. Hiai
    • Organizer
      ARCC Workshop: Positivity, Graphical Models, and Modeling of Complex Multivariate Dependencies
    • Place of Presentation
      パロアルト(米国)
    • Year and Date
      2014-10-15
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Anti Lie-Trotter formula2014

    • Author(s)
      F. Hiai
    • Organizer
      The 19th Conference of the International Linear Algebra Society
    • Place of Presentation
      ソウル(韓国)
    • Year and Date
      2014-08-06
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Anti-norms and superadditivity inequalities on positive matrices and in type II_1 factors2014

    • Author(s)
      F. Hiai
    • Organizer
      16th Workshop: Non-commutative Harmonic Analysis
    • Place of Presentation
      ベドレボ(ポーランド)
    • Year and Date
      2014-07-11
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Different approaches to mutual information in free probability theory2014

    • Author(s)
      F. Hiai
    • Organizer
      Random Matrix Theory: Foundations and Applications
    • Place of Presentation
      クラコフ(ポーランド)
    • Year and Date
      2014-07-04
    • Related Report
      2014 Research-status Report
    • Invited

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Published: 2014-04-04   Modified: 2018-03-22  

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