An approach to free independence based on mutual information

• Project/Area Number: 16K13762
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Nagoya University (2017-2018); Kyushu University (2016)
• Principal Investigator: UEDA Yoshimichi 名古屋大学, 多元数理科学研究科, 教授 (00314724)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 自由独立性 / エントロピー / 相互情報量 / ランダム行列 / 作用素環 / 自由確率論 / 大偏差原理 / 行列値確率過程 / 熱核 / random matrix / process / entropy / free probability / 自由独立 / 情報量

Outline of Final Research Achievements

This research aimed at developing a theory of mutual information in free probability theory to understand the so-called free independence, the central concept of free probability theory. Two kinds of free probabilistic mutual information have been known so far; one is due to Voiculescu in 1999 and the other is due to Hiai, Miyamoto and myself in 2009. However, only a little about them, including their relationship, is known. In this research program, I first wrote a short note indicating an issue that I think is of importance, and then studied the question of large deviations for the matrix liberation process that I introduced as a key to develop the theory of free probabilistic mutual information.

Academic Significance and Societal Importance of the Research Achievements

自由確率論はある作用素環の理解を目指して開発された純粋数学理論であったが，ランダム行列理論との深い関連が明らかになり，作用素環論に留まらず，移動体通信等の工学分野の研究でも利用されるようになった．このことから，直接的な社会的意義は既に明確である．他方で，自由確率論の最大の特性は自由独立性の概念であり，応用する際も何らかの形で自由独立性の概念に基づいていることを考えると，その基礎理論の構築の価値は明白であろう．しかし，まだその理解は初歩的レベルに留まっていると言わざるを得ない．自由独立性の深い理解を目指し，相互情報量の理論の構築を試みることの価値は明白である．

Research Products

• [Journal Article] Matrix liberation process I: Large deviation upper bound and almost sure convergence (2019)
  • Author(s): Yoshimichi Ueda
  • Journal Title: Journal of Theoretical probability Volume: 32 Issue: 2 Pages: 806-847
  • DOI: 10.1007/s10959-018-0819-z
  • NAID: 120006711423
  • Peer Reviewed / Open Access
• [Journal Article] Matrix Liberation Process I: Large Deviation Upper Bound and Almost Sure Convergence (2018)
  • Author(s): Yoshimichi Ueda
  • Journal Title: J. Theor. Probat. Volume: 印刷中
  • NAID: 120006711423
  • Peer Reviewed
• [Journal Article] A remark on orbital free entropy (2017)
  • Author(s): Yoshimichi Ueda
  • Journal Title: Arch. Math. Volume: 108 Pages: 629-638
  • NAID: 120006714549
  • Peer Reviewed
• [Presentation] Matrix liberation process (2019)
  • Author(s): Yoshimichi Ueda
  • Organizer: Free probability: the theory, its extensions, CRM Montreal
  • Invited
• [Presentation] Matrix liberation process and a free probability question (2018)
  • Author(s): Yoshimichi Ueda
  • Organizer: Random Matrices and their applications, Kyoto University
  • Invited
• [Presentation] Matrix liberation process and free probabilistic mutual information (2018)
  • Author(s): Yoshimichi Ueda
  • Organizer: Seminar talk, Mathematical Institute, Harbin Institute of Technology
  • Invited
• [Presentation] Analysis of pairs of projections in free probability (2018)
  • Author(s): Yoshimichi Ueda
  • Organizer: Noncommutative probability and related topics, Hokkaido University
  • Invited
• [Presentation] Matrix liberation process (2017)
  • Author(s): Yoshimichi Ueda
  • Organizer: Workshop "Probabilistic techniques and Quantum Information Theory”, IHP
• [Remarks] 植田好道のホームページ
  • URL: http://www.math.nagoya-u.ac.jp/~ueda/index-j.html
• [Remarks]
  • URL: http://www.math.nagoya-u.ac.jp/~ueda/index.html
• [Remarks] Publications by Y. UEDA
  • URL: http://www2.math.kyushu-u.ac.jp/~ueda/publ.html