Exploring Information Geometrical Methods in Noncommutative Probability Theory

• Project/Area Number: 22340019
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Osaka University
• Principal Investigator: Fujiwara Akio 大阪大学, 理学(系)研究科(研究院), 教授 (30251359)
• Co-Investigator(Kenkyū-buntansha): NAGAOKA Hiroshi 電気通信大学, 情報システム学研究科, 教授 (80192235)
• Co-Investigator(Renkei-kenkyūsha): OHARA Atsumi 福井大学, 工学研究科, 教授 (90221168) ; MABUCHI Toshiki 大阪大学, 理学研究科, 名誉教授 (80116102)
• Project Period (FY): 2010-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥16,120,000 (Direct Cost: ¥12,400,000、Indirect Cost: ¥3,720,000); Fiscal Year 2014: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2013: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2012: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2011: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2010: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
• Keywords: 情報幾何学 / 量子確率空間 / 量子統計学 / 非可換確率論 / Markov不変性 / 量子仮説検定 / 局所漸近正規性 / 単調計量 / 双対平坦 / 適応的量子推定 / Finsler計量 / 繰り込み群 / 撹乱パラメタ

Outline of Final Research Achievements

Aiming at establishing quantum information geometrical methods in noncommutative statistics, we studied the following: 1) Information geometry of renormalization groups, 2) Information geometry of quantum nuisance parameters, 3) Finslerian information geometry, 4) Flattenability of quantum monotone metrics, 5) Experimental verification of adaptive quantum state estimation theory, 6) Quantum local asymptotic normality, 7) Chentsov's theorem revisited, 8) Information geometrical structure induced from sandwiched quantum Renyi relative entropy.

Research Products

• [Journal Article] Quantum local asymptotic normality based on a new quantum likelihood ratio (2013)
  • Author(s): Koichi Yamagata, Akio Fujiwara, Richard D. Gill
  • Journal Title: Annals of Statistics Volume: 41 Issue: 4 Pages: 2197-2217
  • DOI: 10.1214/13-aos1147
  • Peer Reviewed
• [Journal Article] Experimental demonstration of adaptive quantum state estimation (2012)
  • Author(s): R.Okamoto, M.Iefuji, S.Oyama, K.Yamagata, H.Imai, A.Fujiwara and S.Takeuchi
  • Journal Title: Phys. Rev. Lett Volume: Vol.109 Issue: 13 Pages: 130404-130404
  • DOI: 10.1103/physrevlett.109.130404
  • NAID: 120004883866
  • Peer Reviewed
• [Presentation] Amari-Chentsov のパラドクス (2016)
  • Author(s): 藤原彰夫
  • Organizer: 数理と工学の接続と調和
  • Place of Presentation: KKRホテル熱海
  • Year and Date: 2016-02-27
• [Presentation] Experimental Demonstration of Adaptive Quantum State Estimation (2015)
  • Author(s): Akio Fujiwara
  • Organizer: Quantum Information Workshop
  • Place of Presentation: National Institute of Informatics
  • Year and Date: 2015-02-17
  • Int'l Joint Research / Invited
• [Presentation] Geometry of quantum channel manifolds (2010)
  • Author(s): 藤原彰夫
  • Organizer: Information Geometry and its Applications III
  • Place of Presentation: University of Leipzig (Germany)(招待)
  • Year and Date: 2010-08-06
• [Book] 情報幾何学の基礎 (2015)
  • Author(s): 藤原彰夫
  • Total Pages: 223
  • Publisher: 牧野書店