Study of inequalities and variational principle in free probability theory

• Project/Area Number: 21540208
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Tohoku University
• Principal Investigator: HIAI Fumio 東北大学, 大学院・情報科学研究科, 名誉教授 (30092571)
• Co-Investigator(Renkei-kenkyūsha): UEDA Yoshimichi 九州大学, 大学院・数理学研究院, 准教授 (00314724)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2009: ¥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 / 量子情報 / 自由確率論 / 行列 / 量子確率 / 確率論 / 作用素環 / 作用素論 / 量子確率論 / 量子情報理論 / ランダム行列

Research Abstract

In the course of this research project we obtained various results in the three subjects of free probability, quantum probability and operator/matrix analysis. We studied free probability theory related to free entropy and free Fisher information. In quantum probability and quantum information, our study was directed to quantum relative entropy, quantum hypothesis testing and monotone Riemannian metrics. Several results in matrix analysis were also obtained concerning operator monotone and convex functions, matrix trace and norm functions, operator means and symmetric (anti) norms, etc.

Research Products

• [Journal Article] Families of completely positive maps associated with monotone metrics, Linear Algebra Appl (2013)
  • Author(s): F. Hiai, H. Kosaki, D. Petz and M. B. Ruskai
  • Journal Title: to appear
  • URL: http://www.journals.elsevier.com/linear-algebra-and-its-applications/
• [Journal Article] Higher order extension of Loewner's theory: Operator k-tone functions (2013)
  • Author(s): U. Franz
  • Journal Title: Trans. Amer. Math. Soc. Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Concavity of certain matrix trace and norm functions (2013)
  • Author(s): F. Hiai
  • Journal Title: Linear Algebra Appl. Volume: 印刷中
  • Peer Reviewed
• [Journal Article] On matrix inequalities between the power means: counterexamples (2013)
  • Author(s): K. M. R. Audenaert
  • Journal Title: Linear Algebra Appl. Volume: 印刷中
  • Peer Reviewed
• [Journal Article] From quasi-entropy to various quantum information quantities (2012)
  • Author(s): F. Hiai and D. Petz
  • Journal Title: Publ. Res. Inst. Math. Sci Volume: 48 Issue: 3 Pages: 525-542
  • DOI: 10.2977/prims/79
• [Journal Article] Riemannian metrics on positive definite matrices related to means (2012)
  • Author(s): F. Hiai and D. Petz
  • Journal Title: II, Linear Algebra Appl Volume: 436 Issue: 7 Pages: 2117-2136
  • DOI: 10.1016/j.laa.2011.10.029
  • Peer Reviewed
• [Journal Article] Loewner matrices of matrix convex and monotone functions (2012)
  • Author(s): F. Hiai, T. Sano
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 64 Issue: 2 Pages: 343-364
  • DOI: 10.2969/jmsj/06420343
  • NAID: 10030740187
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] From quasi-entropy to various quantum information quantities (2012)
  • Author(s): F.Hiai
  • Journal Title: Publ.Res.Inst.Math.Sci. Volume: 48(印刷中)
  • Peer Reviewed
• [Journal Article] Quantum f-divergences and error correction (2011)
  • Author(s): F. Hiai, M. Mosonyi, D. Petz and C. Beny
  • Journal Title: Rev. Math. Phys Volume: 23 Issue: 07 Pages: 691-747
  • DOI: 10.1142/s0129055x11004412
  • Peer Reviewed
• [Journal Article] Norm and anti-norm inequalities for positive semi-definite matrices (2011)
  • Author(s): J.-C. Bourin and F. Hiai
  • Journal Title: Internat. J. Math Volume: 22 Issue: 08 Pages: 1121-1138
  • DOI: 10.1142/s0129167x1100715x
• [Journal Article] On the quantum Renyi relative entropies and related capacity formulas (2011)
  • Author(s): M. Mosonyi and F. Hiai
  • Journal Title: IEEE Trans. Inform. Theory Volume: 57 Issue: 4 Pages: 2474-2487
  • DOI: 10.1109/tit.2011.2110050
  • Peer Reviewed
• [Journal Article] Operator log-convex functions and operator means (2011)
  • Author(s): T. Ando and F. Hiai
  • Journal Title: Math. Ann Volume: 350 Issue: 3 Pages: 611-630
  • DOI: 10.1007/s00208-010-0577-4
  • Peer Reviewed
• [Journal Article] On the quantum Renyi relative entropies and related capacity formulas (2011)
  • Author(s): M.Mosonyi
  • Journal Title: IEEE Trans.Inform.Theory Volume: 未定(印刷中)
  • Peer Reviewed
• [Journal Article] Loewner matrices of matrix convex and monotone functions (2011)
  • Author(s): F.Hiai
  • Journal Title: J.Math.Soc.Japan Volume: 未定(印刷中)
  • NAID: 10030740187
  • Peer Reviewed
• [Journal Article] Norm and anti-norm inequalities for positive semi-definite matrices (2011)
  • Author(s): J.-C.Bonrin
  • Journal Title: Internat.J.Math. Volume: 未定(印刷中)
  • Peer Reviewed
• [Journal Article] Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization (2010)
  • Author(s): F. Hiai
  • Journal Title: Interdisciplinary Information Sciences Volume: 16 Issue: 2 Pages: 139-246
  • DOI: 10.4036/iis.2010.139
  • NAID: 130000323465
  • ISSN: 1340-9050, 1347-6157
• [Journal Article] Strongly subadditive functions (2010)
  • Author(s): K. Audenaert, F. Hiai and D. Petz
  • Journal Title: Acta Math. Hungar Volume: 128 Issue: 4 Pages: 386-394
  • DOI: 10.1007/s10474-010-9222-7
• [Journal Article] A new approach to mutual information. II (2010)
  • Author(s): F.Hiai
  • Journal Title: Banach Center Publications Volume: 89 Pages: 143-163
  • Peer Reviewed
• [Journal Article] Strongly subadditive functions (2010)
  • Author(s): K.Audenaer
  • Journal Title: Acta Math.Hungar. Volume: 128 Pages: 386-394
  • Peer Reviewed
• [Journal Article] Matrix Analysis : Matrix Monotone Functions, Matrix Means, and Majorization (GSIS selected lectures) (2010)
  • Author(s): F.Hiai
  • Journal Title: Interdisciplinary Information Sciences Volume: 16 Pages: 139-248
  • NAID: 110009795065
  • Peer Reviewed
• [Journal Article] Quantum hypothesis testing with group symmetry (2009)
  • Author(s): F. Hiai, M. Mosonyi and M. Hayashi
  • Journal Title: J. Math. Phys Volume: 50 Issue: 10 Pages: 1-31
  • DOI: 10.1063/1.3234186
• [Journal Article] Monotonicity for entrywise functions of matrices (2009)
  • Author(s): F. Hiai
  • Journal Title: Linear Algebra Appl Volume: 431 Issue: 8 Pages: 1125-1146
  • DOI: 10.1016/j.laa.2009.04.001
• [Journal Article] Riemannian metrics on positive definite matrices related to means (2009)
  • Author(s): F. Hiai and D. Petz
  • Journal Title: Linear Algebra Appl Volume: 430 Issue: 11-12 Pages: 3105-3130
  • DOI: 10.1016/j.laa.2009.01.025
• [Journal Article] Quantum covariance, quantum Fisher information and the uncertainty principle (2009)
  • Author(s): P. Gibilisco, F. Hiai and D. Petz
  • Journal Title: IEEE Trans. Inform. Theory Volume: 55 Issue: 1 Pages: 439-443
  • DOI: 10.1109/tit.2008.2008142
• [Journal Article] Orbital approach to microstate free entropy (2009)
  • Author(s): F. Hiai, T. Miyamoto and Y. Ueda
  • Journal Title: Internat. J. Math Volume: 20 Issue: 02 Pages: 227-273
  • DOI: 10.1142/s0129167x09005261
  • NAID: 120006714544
• [Journal Article] A log-Sobolev type inequality for free entropy of two projections (2009)
  • Author(s): F. Hiai and Y. Ueda
  • Journal Title: Ann. Inst. H. Poincare Probab. Statist Volume: 45 Issue: 1 Pages: 239-249
  • DOI: 10.1214/08-aihp164
  • NAID: 120006714550
• [Journal Article] A log-Sobolev type inequality for free entropy of two projections (2009)
  • Author(s): F.Hiai
  • Journal Title: Ann.Inst.H.Poincare Probab.Statist. 45 Pages: 239-249
  • NAID: 120006714550
  • Peer Reviewed
• [Journal Article] Orbital approach to microstate free entropy (2009)
  • Author(s): F.Hiai
  • Journal Title: Internat.J.Math. 20 Pages: 227-273
  • NAID: 120006714544
  • Peer Reviewed
• [Journal Article] Quantum covariance, quantum Fisher information and the uncertainty principle (2009)
  • Author(s): P.Gibilisco
  • Journal Title: IEEE Trans.Inform.Theory 55 Pages: 439-443
  • Peer Reviewed
• [Journal Article] Riemannian metrics on positive definite matrices related to means (2009)
  • Author(s): F.Hiai
  • Journal Title: Linear Algebra Appl. 430 Pages: 3105-3130
  • Peer Reviewed
• [Journal Article] Monotonicity for entrywise functions of matrices (2009)
  • Author(s): F.Hiai
  • Journal Title: Linear Algebra Appl. 431 Pages: 1125-1146
  • Peer Reviewed
• [Journal Article] Quantum hypothesis testing with group symmetry (2009)
  • Author(s): F.Hiai
  • Journal Title: J.Math.Phys. 50
  • Peer Reviewed
• [Journal Article] On matrix inequalities between the power means: counterexamples
  • Author(s): K. M. R. Audenaert and F. Hiai
  • Journal Title: Linear Algebra Appl., to appear
  • URL: http://dx.doi.org/10.1016/j.laa.2013.04.012
• [Journal Article] Concavity of certain matrix trace and norm functions
  • Author(s): F. Hiai
  • Journal Title: Linear AlgebraAppl., to appear
  • URL: http://dx.doi.org/10.1016/j.laa.2013.04.020
• [Journal Article] Ricard, Higher order extension of Lowner's theory: Operator k-tone functions
  • Author(s): U. Franz, F. Hiai and E
  • Journal Title: Trans. Amer. Math. Soc., to appear
  • URL: http://www.ams.org/jourcgi/jrnl_toolbar_nav/tran_all
• [Journal Article] A new approach to mutual information. II
  • Author(s): F.Hiai
  • Journal Title: Banach Center Publications (未定, 印刷中)
  • Peer Reviewed
• [Presentation] Concavity of certain matrix trace and norm functions,"Matrices and Operators" (2012)
  • Author(s): F. Hiai
  • Organizer: Conference in Honour of Sixtieth Birthday of Rajendra Bhatia
  • Place of Presentation: Indian Institute of Science, Bangarole, India
  • Year and Date: 2012-12-29
• [Presentation] Analytic functional calculus of operator convex/monotone functions (2012)
  • Author(s): F. Hiai
  • Organizer: The 2012 KOTAC International Conference: Operator Theory and Its Applications
  • Place of Presentation: Chungbuk National University, Cheongju, Korea
  • Year and Date: 2012-06-22
• [Presentation] Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited, International Workshop on Anomalous Statistics (2012)
  • Author(s): F. Hiai
  • Organizer: Generalized Entropies, and Information Geometry
  • Place of Presentation: Nara Women's University, Nara, Japan
  • Year and Date: 2012-03-10
• [Presentation] Concavity of certain matrix trace and norm functions (2012)
  • Author(s): F. Hiai
  • Organizer: Matrices and Operators, Conference in Honour of Sixtieth Birthday of Rajendra Bhatia
  • Place of Presentation: Indian Institute of Science, Bangarole, India
  • Invited
• [Presentation] Convexity of quasi-entropy type functions : Lieb's and Ando's convexity theorems revisited (2012)
  • Author(s): F.Hiai
  • Organizer: International Workshop on Anomalous Statistics, Generalized Entropies, and Information Geometry
  • Place of Presentation: Nara Women's University, Nara, Japan(招待講演)
• [Presentation] Toward the final version of Lieb-Ando concavity (2011)
  • Author(s): 日合 文雄
  • Organizer: 研究集会「スペクトル,数域などの作用素の幾何的特性量を用いた作用素の構造研究」
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2011-11-15
• [Presentation] Toward the final version of Lieb-Ando concavity (2011)
  • Author(s): 日合文雄
  • Organizer: 研究集会「スペクトル,数域などの作用素の幾何的特性量を用いた作用素の構造研究」
  • Place of Presentation: 京都大学,数理解析研究所(招待講演)
  • Year and Date: 2011-11-15
• [Presentation] Higher order extension of Lowner's theory: Operator k-tone functions (2011)
  • Author(s): 日合 文雄
  • Organizer: 作用素論・作用素環論研究集会
  • Place of Presentation: 琉球大学
  • Year and Date: 2011-11-04
• [Presentation] Higher order extension of L"owner's theory : Operator k-tone functions (2011)
  • Author(s): 日合文雄
  • Organizer: 作用素論・作用素環論研究集会
  • Place of Presentation: 琉球大学,50周年記念館(招待講演)
  • Year and Date: 2011-11-04
• [Presentation] Jensen and Minkowski type inequalities for operator means (joint work with J.-C. Bourin) (2011)
  • Author(s): F. Hiai
  • Organizer: 14th Workshop"Non-commutative Harmonic Analysis"
  • Place of Presentation: Bedlewo, Poland
  • Year and Date: 2011-09-30
• [Presentation] Free energy density in one-dimensional spin chains (2011)
  • Author(s): F. Hiai
  • Organizer: Summer school on "Non-equilibrium statistical mechanics"
  • Place of Presentation: Centre de recherches mathematiques, Montreal, Canada
  • Year and Date: 2011-07-20
• [Presentation] Free energy density in one-dimensional spin chains (2011)
  • Author(s): F.Hiai
  • Organizer: Summer school on "Non-equilibrium statistical mechanics"
  • Place of Presentation: Centre de recherches math'ematiques, Montr'eal, Canada(招待講演)
  • Year and Date: 2011-07-20
• [Presentation] Generalizing Minkowski's determinantal inequality (joint work with J.-C. Bourin), International Conference (2011)
  • Author(s): F. Hiai
  • Organizer: Product Systems and Independence in Quantum Dynamics
  • Place of Presentation: Alfried Krupp Wissenschaftskolleg, Greifswald, Germany
  • Year and Date: 2011-03-14
• [Presentation] Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited (2011)
  • Author(s): F. Hiai
  • Organizer: The 6th Jikji Workshop: Infinite Dimensional Analysis and Quantum Probability
  • Place of Presentation: National Institute for Mathematical Sciences, Daejeon, Korea
  • Year and Date: 2011-01-10
• [Presentation] Jensen and Minkowski type inequalities for operator means (joint work with J.-C.Bourin) (2011)
  • Author(s): F.Hiai
  • Organizer: 14th Workshop "Non-commutative Harmonic Analysis"
  • Place of Presentation: The Mathematical Research and Conference Center, Bedlewo, Poland(招待講演)
• [Presentation] Convexity of quasi-entropy type functions : Lieb's and Ando's convexity theorems revisited (2011)
  • Author(s): F.Hiai
  • Organizer: The 6th Jikji Workshop : Irfiinite Dimensional Analysis and Quantum Probability
  • Place of Presentation: National Institute for Mathematical Sciences, Daejeon, Korea
• [Presentation] Generalizing Minkowski's determinantal inequality (joint work with J.-C.Bourin) (2011)
  • Author(s): F.Hiai
  • Organizer: International Conference "Product Systems and Independence in Quantum Dynamics"
  • Place of Presentation: Alfried Krupp Wissenschaftskolleg, Greifswald, Germany
• [Presentation] Operator log-convexity and its applications (2010)
  • Author(s): F. Hiai
  • Organizer: Program: Quantum Information Theory
  • Place of Presentation: Institut Mittag-Leffler, Djursholm, Sweden
  • Year and Date: 2010-10-26
• [Presentation] Operator log-convexity and its applications (2010)
  • Author(s): F.Hiai
  • Organizer: Program : Quantum Information Theory
  • Place of Presentation: Institut Mittag-Lefler, Djursholm, Sweden
  • Year and Date: 2010-10-26
• [Presentation] Riemannian metrics on positive definite matrices related to means (joint work with Denes Petz) (2010)
  • Author(s): F. Hiai
  • Organizer: Information Geometry and its Applications III
  • Place of Presentation: Max Planck Institut, Leipzig, Germany.
  • Year and Date: 2010-08-04
• [Presentation] Operator log-convex functions and operator means (joint work with T. Ando) (2010)
  • Author(s): F. Hiai
  • Organizer: the 16th Conference of the International Linear Algebra Society
  • Place of Presentation: Palazzo dei Congress, Pisa,Italy
  • Year and Date: 2010-06-22
• [Presentation] Operator log-convex functions and operator means (joint work with T.Ando) (2010)
  • Author(s): F.Hiai
  • Organizer: The 16th Conference of the International Linear Algebra Society
  • Place of Presentation: Palazzo dei Congress, Pisa, Italy
• [Presentation] Riemannian metrics on positive definite matrices related to means (joint work with Denes Petz) (2010)
  • Author(s): F.Hiai
  • Organizer: Information Geometry and its Applications III
  • Place of Presentation: Max-Planck-Institut, Leipzig, Germany
• [Presentation] 正定値行列の空間における平均と関連するRiemann計量 (2009)
  • Author(s): 日合文雄
  • Organizer: 研究集会「作用素論における非可換解析学の展望」
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2009-10-29
• [Presentation] Riemannian metrics on positive definite matrices related to means (joint work with D.Petz) (2009)
  • Author(s): F. Hiai
  • Organizer: 12th Workshop "Non-commutative Harmonic Analysis"
  • Place of Presentation: Bedlewo, Poland
  • Year and Date: 2009-08-21
• [Presentation] Riemannian metrics on positive definite matrices related to means(joint work with D. Petz) (2009)
  • Author(s): F.Hiai
  • Organizer: 12th Workshop "Non-commutative Harmonic Analysis"
  • Place of Presentation: Mathematical Research and Conference Center, Bedlewo, Poland
• [Presentation] 平均と関連する正定値行列の空間におけるリーマン計量 (2009)
  • Author(s): 日合文雄
  • Organizer: 研究集会「作用素論における非可換解析学の展望」
  • Place of Presentation: 京都大学理学研究科
• [Remarks]
  • URL: http://db.tohoku.ac.jp/whois/detail/340f360473ce9efb66b08f86f84572e5.html
• [Remarks]
  • URL: http://db.tohoku.ac.jp/whois/detail/340f360473ce9efb66b08f86f84572e5.html