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Statistical Estimation Theory for Quantum Channels

Research Project

Project/Area Number 15340031
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionOsaka University

Principal Investigator

FUJIWARA Akio  Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (30251359)

Co-Investigator(Kenkyū-buntansha) SUGIMOTO Mitsuru  Osaka University, Graduate School of Science, Associate Professor, 理学研究科, 助教授 (60196756)
KONNO Kazuhiro  Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (10186869)
SUZUKU Joe  Osaka University, Graduate School of Science, Associate Professor, 理学研究科, 助教授 (50216397)
MABUCHI Toshiki  Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (80116102)
WATANABE Takao  Osaka University, Graduate School of Science, Professor, 理学研究科, 教授 (30201198)
山崎 洋平  大阪大学, 理学研究科, 助教授 (00093477)
眞鍋 昭治郎  大阪大学, 大学院・理学研究科, 助教授 (20028260)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥8,700,000 (Direct Cost: ¥8,700,000)
Fiscal Year 2005: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2004: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2003: ¥3,600,000 (Direct Cost: ¥3,600,000)
Keywordsquantum channel / complete positivity / information geometry / adaptive estimation / Cramer-Rao inequality / martingale / representation theory / asymptotics / パラメータ推定 / 量子情報 / 量子幾何 / エンタングルド状態 / Pauli通信路(パウリ通信路) / 非可換統計学 / 量子Fisher情報量
Research Abstract

This research addresses the problem of estimating an unknown quantum channel, based on noncommutative statistics and quantum information geometry. Main results are summarized as follows :
1. Estimation theory for generalized Pauli channels, generalized amplitude-damping channel, SU(d) channel, etc. :
These channels are the standard ones often used in quantum information theory. Based on noncommutative statistics, quantum information geometry, representation theory, as well as methods of experimental mathematics on a computer, we have obtained the optimal estimation scheme for each of those quantum channels. We also have clarified the underlining differential geometrical mechanism, in particular, the dilation/collapse of quantum statistical manifolds of output quantum states with respect to the quantum entanglement and the degree of extensions, behind the optimality of those estimation schemes.
2. Asymptotic theory of adaptive estimation schemes based on the locally unbiased estimators :
The notion of locally unbiased estimators, advocated first by Holevo, has been the target of criticism because the optimal estimation scheme depends on the unknown parameter itself. In order to surmount this difficulty, Nagaoka advocated an adaptive estimation scheme which make use of maximal likelihood estimators as a tentative estimator. However, due to the mathematical difficulty, the analysis of its asymptotic property has been left untouched. In this research, we have proved the strong consistency and the asymptotic efficiency of Nagaoka's adaptive estimation scheme based on the martingale theory.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (17 results)

All 2005 2004 2003 Other

All Journal Article (16 results) Publications (1 results)

  • [Journal Article] Information geometry of quantum channel estimation2005

    • Author(s)
      Akio Fujiwara
    • Journal Title

      Proc. 2^<nd> International Symposium on Information Geometry and Its Applications

      Pages: 170-170

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Information geometry of quantum channel estimation2005

    • Author(s)
      A.Fujiwara
    • Journal Title

      Proc.2^<nd> International Symposium on Information Geometry and Its Applications

      Pages: 170-170

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Estimation of a generalized amplitude-damping channel2004

    • Author(s)
      Akio Fujiwara
    • Journal Title

      Physical Review A 70

      Pages: 12317-12317

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Statistical estimation of a quantum operation2004

    • Author(s)
      Akio Fujiwara
    • Journal Title

      Quantum Information, Statistics, Probability

      Pages: 78-78

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Statistical estimation of a quantum operation2004

    • Author(s)
      Akio Fujiwara
    • Journal Title

      Quantum Information and Computation 4

      Pages: 479-479

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Estimation of a generalized amplitude-damping channel2004

    • Author(s)
      A.Fujiwara
    • Journal Title

      Physical Review A vol.70

      Pages: 12317-12317

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Statistical estimation of a quantum operation2004

    • Author(s)
      A.Fujiwara
    • Journal Title

      Quantum Information Statistics, Probability

      Pages: 78-78

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Statistical estimation of a quantum operation2004

    • Author(s)
      A.Fujiwara
    • Journal Title

      Quantum Information and Computation vol.4

      Pages: 479-479

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Estimation of a generalized amplitude-damping channel2004

    • Author(s)
      Akio FUJIWARA
    • Journal Title

      Physical Review A 70

      Pages: 12317-12317

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Statistical estimation of a quantum operation2004

    • Author(s)
      Akio FUJIWARA
    • Journal Title

      Quantum Information, Statistics, Probability

      Pages: 78-89

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Quantum parameter estimation of a generalized Pauli channel2003

    • Author(s)
      Akio Fujiwara
    • Journal Title

      Journal of Physics A : Mathematical and General 36

      Pages: 8093-8093

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Quantum parameter estimation of a generalized Pauli channel2003

    • Author(s)
      A.Fujiwara, H.Imai
    • Journal Title

      Journal of Physics A : Mathematical and General vol.36

      Pages: 8093-8093

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Entanglement-assisted estimation of quantum channels

    • Author(s)
      Akio Fujiwara
    • Journal Title

      Proc. 8^<th> International Symposium on Foundations of Quantum Mechanics in the Light of New Technology (In press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Entanglement-assisted estimation of quantum channels

    • Author(s)
      A.Fujiwara
    • Journal Title

      Proc.8^<th> International Symposium on Foundations of Quantum Mechanics in the Light of New Technology (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Entanglement-assisted estimation of quantum channels

    • Author(s)
      Akio FUJIWARA
    • Journal Title

      Proc.8^<th> International Symposium on Foundations of Quantum Mechanics in the Light of New Technology (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Information geometry of quantum channel estimation

    • Author(s)
      Akio FUJIWARA
    • Journal Title

      Proc.2^<nd> International Symposium on Information Geometry and its Applications (in press)

    • Related Report
      2005 Annual Research Report
  • [Publications] A.Fujiwara, H.Imai: "Quantum parameter estimation of a generalized Pauli channel"Journal of Physics A : Mathematical and General. vol.36. 8093-8103 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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