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Random Matrix Theory and applications to Quantum Information Theory

Research Project

Project/Area Number 17H04823
Research Category

Grant-in-Aid for Young Scientists (A)

Allocation TypeSingle-year Grants
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

Collins Benoit  京都大学, 理学研究科, 教授 (20721418)

Project Period (FY) 2017-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥24,050,000 (Direct Cost: ¥18,500,000、Indirect Cost: ¥5,550,000)
Fiscal Year 2020: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2019: ¥5,850,000 (Direct Cost: ¥4,500,000、Indirect Cost: ¥1,350,000)
Fiscal Year 2018: ¥8,060,000 (Direct Cost: ¥6,200,000、Indirect Cost: ¥1,860,000)
Fiscal Year 2017: ¥5,460,000 (Direct Cost: ¥4,200,000、Indirect Cost: ¥1,260,000)
KeywordsFree probability / Random Matrices / free probability / operator norm / ランダム行列理論 / ランダム行列論 / 自由確率論 / 量子情報 / Random Matrix / Quantum Information / ランダム行列 / random matrix theory
Outline of Final Research Achievements

With Bordenave, we developed the theory of operator valued non-backtracking and showed strong asymptotically freeness for random permutations, which led to the solution of the Alon conjecture, an important unsolved problem for 20 years. Together with Bordenave again, we proved strong asymptotic freeness for tensors of unitary random matrices. With Guionnet and Parraud, we gave a new quantitative proof of the results of the original paper by Haagerup and Thorbjornsen. Based on this as an application to quantum information theory, in collaboration with Parraud, we showed the bound of dimension in which the MOE additivity violation occur.

Academic Significance and Societal Importance of the Research Achievements

量子物理に強い動機を持つランダム行列理論は理論的な興味だけでなく多くの科学で応用がなされている。量子物理だけでなく、リーマン予想や代数幾何のような数学の基礎を成すような分野から数理ファイナンスや情報科学、量子計算、学習理論などの応用分野まで幅広い分野にその応用を持っている。実際本研究課題でも量子コンピュータの開発などで注目されている量子情報理論で重要なMOEの加法性の破れを定量的に考察することに成功した。最近では深層学習をはじめとする機械学習分野でもランダム行列の活用が始まっており、本研究に関わる結果は今後の社会の基礎数理としての期待があると考えている。

Report

(5 results)
  • 2022 Final Research Report ( PDF )
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • Research Products

    (54 results)

All 2023 2022 2020 2019 2018 2017 Other

All Int'l Joint Research (1 results) Journal Article (20 results) (of which Int'l Joint Research: 15 results,  Peer Reviewed: 19 results,  Open Access: 8 results) Presentation (31 results) (of which Int'l Joint Research: 8 results,  Invited: 31 results) Remarks (2 results)

  • [Int'l Joint Research] Aix-Marseille University CNRS(フランス)

    • Related Report
      2020 Annual Research Report
  • [Journal Article] Asymptotic freeness of layerwise Jacobians caused by invariance of multilayer perceptron: the Haar orthogonal case2023

    • Author(s)
      Collins, Beno\^{i}t and Hayase, Tomohiro
    • Journal Title

      Communications in Mathematical Physics

      Volume: 397 Issue: 1 Pages: 85-109

    • DOI

      10.1007/s00220-022-04441-7

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The spectrum of local random Hamiltonians2023

    • Author(s)
      Collins, B. and Yin, Z. and Zhao, L. and Zhong, P.
    • Journal Title

      Journal of Physics. A. Mathematical and Theoretical

      Volume: 56 Issue: 3 Pages: 035201-28

    • DOI

      10.1088/1751-8121/acb4c8

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A metric characterization of freeness2022

    • Author(s)
      Cadilhac, L. and Collins, B.
    • Journal Title

      Journal of Functional Analysis

      Volume: 283 Issue: 5 Pages: 109562-14

    • DOI

      10.1016/j.jfa.2022.109562

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The Weingarten Calculus2022

    • Author(s)
      B. Collins Benoit, S. Matsumoto Sho, J. Novak
    • Journal Title

      Notices of the American Mathematical Society

      Volume: 69 Issue: 05 Pages: 734-745

    • DOI

      10.1090/noti2474

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices2022

    • Author(s)
      Collins, Benoit and Guionnet, Alice and Parraud, Felix
    • Journal Title

      Cambridge Journal of Mathematics

      Volume: 10

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On spectral distribution of sample covariance matrices from large dimensional and large k-fold tensor products2022

    • Author(s)
      Collins, Benoit and Yao, Jianfeng and Yuan, Wangjun
    • Journal Title

      Electronic Journal of Probability

      Volume: 27 Issue: none Pages: 102-18

    • DOI

      10.1214/22-ejp825

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Additivity violation of the regularized minimum output entropy2022

    • Author(s)
      Collins, Benoit and Youn, Sang-Gyun
    • Journal Title

      Documenta Mathematica

      Volume: 27 Pages: 1299-1320

    • DOI

      10.4171/dm/898

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Concentration estimates for random subspaces of a tensor product and application to quantum information theory2022

    • Author(s)
      Collins, Benoit and Parraud, Felix
    • Journal Title

      Journal of Mathematical Physics

      Volume: 63 Issue: 10 Pages: 102202-13

    • DOI

      10.1063/5.0073837

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Entanglement and the Temperley-Lieb category2020

    • Author(s)
      Brannan Michael、Collins Benoit
    • Journal Title

      Contemp. Math.

      Volume: 747 Pages: 27-50

    • DOI

      10.1090/conm/747/15037

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Eigenvalues of random lifts and polynomials of random permutation matrices2019

    • Author(s)
      Bordenave、Collins
    • Journal Title

      Annals of Mathematics

      Volume: 190 Issue: 3 Pages: 811-875

    • DOI

      10.4007/annals.2019.190.3.3

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The PPT square conjecture holds generically for some classes of independent states2018

    • Author(s)
      Collins Benoit、Yin Zhi、Zhong Ping
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 51 Issue: 42 Pages: 425301-425301

    • DOI

      10.1088/1751-8121/aadd52

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On a family of linear maps fromMn(C)toMn2(C)2018

    • Author(s)
      Collins Benoit、Osaka Hiroyuki、Sapra Gunjan
    • Journal Title

      Linear Algebra and its Applications

      Volume: 555 Pages: 398-411

    • DOI

      10.1016/j.laa.2018.06.011

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Free probability for purely discrete eigenvalues of random matrices2018

    • Author(s)
      COLLINS Benoit、HASEBE Takahiro、SAKUMA Noriyoshi
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 70 Issue: 3 Pages: 1111-1150

    • DOI

      10.2969/jmsj/77147714

    • NAID

      130007420328

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Semiclassical asymptotics of $$\mathbf {GL}_N({\mathbb {C}})$$GLN(C) tensor products and quantum random matrices2018

    • Author(s)
      Collins Benoit、Novak Jonathan、Sniady Piotr
    • Journal Title

      Selecta Mathematica

      Volume: 24 Issue: 3 Pages: 2571-2623

    • DOI

      10.1007/s00029-017-0387-6

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Dual bases in Temperley-Lieb algebras, quantum groups, and a question of Jones2018

    • Author(s)
      Brannan Michael、Collins Benoit
    • Journal Title

      Quantum Topology

      Volume: 9 Issue: 4 Pages: 715-748

    • DOI

      10.4171/qt/118

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Random and Free Positive Maps with Applications to Entanglement Detection2017

    • Author(s)
      Benoit Collins、Patrick Hayden、Ion Nechita
    • Journal Title

      International Mathematics Research Notices

      Volume: 3 Pages: 869-894

    • DOI

      10.1093/imrn/rnw054

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Convex set of quantum states with positive partial transpose analysed by hit and run algorithm2017

    • Author(s)
      Szymanski Konrad、Collins Benoit、Szarek Tomasz、Zyczkowski Karol
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 50 Issue: 25 Pages: 255206-255206

    • DOI

      10.1088/1751-8121/aa70f5

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Weingarten calculus via orthogonality relations: new applications2017

    • Author(s)
      Benoit Collins Sho Matsumoto
    • Journal Title

      ALEA. Latin American Journal of Probability and Mathematical Statistics

      Volume: 14 Pages: 631-656

    • Related Report
      2017 Annual Research Report
  • [Journal Article] The spectral edge of unitary Brownian motion2017

    • Author(s)
      Collins Benoit, Dahlqvist Antoine, Kemp Todd
    • Journal Title

      Probability Theory and Related Fields

      Volume: 170 Issue: 1-2 Pages: 49-93

    • DOI

      10.1007/s00440-016-0753-x

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Highly Entangled, Non-random Subspaces of Tensor Products from Quantum Groups2017

    • Author(s)
      Brannan Michael, Collins Benoit
    • Journal Title

      Communications in Mathematical Physics

      Volume: 358 Issue: 3 Pages: 1007-1025

    • DOI

      10.1007/s00220-017-3023-6

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Presentation] New results around the norm of random matrices and operator-valued non-backtracking theory2022

    • Author(s)
      Benoit Collins
    • Organizer
      Geneva MP seminar
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] convergence of the spectrum of random matrices in the context of rational fractions2022

    • Author(s)
      Benoit Collins
    • Organizer
      UCSD seminar
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Matrix integrals in a tensor setup2022

    • Author(s)
      Benoit Collins
    • Organizer
      Bielefeld - Melbourne Zufallsmatrizen
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Properties of random tensor matrices with some applications to Quantum Informa- tion Theory2022

    • Author(s)
      Benoit Collins
    • Organizer
      Functional Analysis Seminar - Institute for Advanced Study in Mathematics of HIT
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Matrix models for cyclic monotone and monotone independences2022

    • Author(s)
      Benoit Collins
    • Organizer
      SIMA 2022 - CIMAT
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Centered Weingarten calculus and centered Wick calculus2022

    • Author(s)
      Benoit Collins
    • Organizer
      Fields Workshop on Noncommutative Geometry, Free Probability Theory and Random Matrix Theory
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Strong convergence2022

    • Author(s)
      Benoit Collins
    • Organizer
      Northwestern Nemmers conference
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Weingarten calculus for tensors2022

    • Author(s)
      Benoit Collins
    • Organizer
      Tensors and applications
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Strong asymptotic freeness for random tensors of unitaries2019

    • Author(s)
      Benoit Collins
    • Organizer
      Probability seminar - Princeton
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Strong asymptotic freeness for random tensors of unitaries2019

    • Author(s)
      Benoit Collins
    • Organizer
      Probability and Mathematical Physics Seminar - NYU
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Regularized minimum output entropy and generalized Haagerup inequalities2019

    • Author(s)
      Benoit Collins
    • Organizer
      Colloquium - Kaohsiung
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Norm estimates for polynomials in random permutations2019

    • Author(s)
      Benoit Collins
    • Organizer
      Probability seminar - Academia Sinica - Taipei
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Strong convergence for random permutations2019

    • Author(s)
      Benoit Collins
    • Organizer
      Colloquium - Melbourne
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Strong convergence for random permutations and random tensors2019

    • Author(s)
      Benoit Collins
    • Organizer
      Japanese-German Open Conference on Stochastic Analysis - Fukuoka
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Strong convergence for random permutations and random tensors2019

    • Author(s)
      Benoit Collins
    • Organizer
      WPI 2019 - Hong Kong. (8) June 2019 - Fields
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Equivariance and application to entanglement detection2019

    • Author(s)
      Benoit Collins
    • Organizer
      Interactions between Noncommutative Analysis and Quantum Information Theory - Harbin
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Strong Asymptotic Freeness for Random Permutations and Random Tensors2019

    • Author(s)
      Benoit Collins
    • Organizer
      NYUS - Shanghai colloquium
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Non commutative probability and the Connes embedding problem2019

    • Author(s)
      Benoit Collins
    • Organizer
      University of Oslo
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] The PPT2 square conjecture holds generically2019

    • Author(s)
      Benoit Collins
    • Organizer
      Chula University Bangkok
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] strong freeness for random tensors2019

    • Author(s)
      Benoit Collins
    • Organizer
      UC Berkeley
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Strong convergence for random permutations2018

    • Author(s)
      Benoit Collins
    • Organizer
      HKUST (Hong Kong)
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Strong convergence for random permutations2018

    • Author(s)
      Benoit Collins
    • Organizer
      UCLA (IPAM)
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Random permutations and quantum information theory2018

    • Author(s)
      Benoit Collins
    • Organizer
      Oberwolfach
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Strong convergence for random permutations2018

    • Author(s)
      Benoit Collins
    • Organizer
      Ottawa
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] The PPT2 square conjecture holds generically2018

    • Author(s)
      Benoit Collins
    • Organizer
      Nagoya
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Random permutations and quantum information theory2018

    • Author(s)
      Benoit Collins
    • Organizer
      Okinawa (OIST)
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Strong convergence for random permutations2018

    • Author(s)
      Benoit Collins
    • Organizer
      Paris 6, France
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Introduction to Weingarten calculus2017

    • Author(s)
      Benoit Collins
    • Organizer
      CIMPA spring school, Settat, Morocco
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Free probability for purely discrete eigenvalues of random matrices2017

    • Author(s)
      Benoit Collins
    • Organizer
      Park City, PCMI, USA
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] MOE estimates for quantum channels arising from random isometries and free probability2017

    • Author(s)
      Benoit Collins
    • Organizer
      IHP, Paris, France
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Strong convergence for random permutations2017

    • Author(s)
      Benoit Collins
    • Organizer
      Orsay, Paris, France
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Remarks] Collins' website

    • URL

      https://www.math.kyoto-u.ac.jp/~collins/

    • Related Report
      2020 Annual Research Report
  • [Remarks] Benoit Collins' website

    • URL

      https://www.math.kyoto-u.ac.jp/~collins/

    • Related Report
      2019 Annual Research Report 2018 Annual Research Report 2017 Annual Research Report

URL: 

Published: 2017-04-28   Modified: 2024-01-30  

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