© 2017

Quantum Symmetries

Metabief, France 2014

  • Uwe Franz
  • Gives an accessible introduction to current research

  • Provides a common point of view on several interconnected, but originally independently developed areas of mathematics

  • Open problems suggest directions for future research

  • Provides a survey of current research topics in functional analysis and its applications to quantum physics

  • Offers a unique blend of uses of quantum symmetries (in quantum groups, non-commutative probability and quantum physics)

  • Covers a variety of different facets of the modern concept of quantum symmetries


Part of the Lecture Notes in Mathematics book series (LNM, volume 2189)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Uwe Franz
    Pages 1-3
  3. Guillaume Aubrun
    Pages 83-114
  4. Back Matter
    Pages 115-119

About this book


Providing an introduction to current research topics in functional analysis and its applications to quantum physics, this book presents three lectures surveying recent progress and open problems.

 A special focus is given to the role of symmetry in non-commutative probability, in the theory of quantum groups, and in quantum physics. The first lecture presents the close connection between distributional symmetries and independence properties. The second introduces many structures (graphs, C*-algebras, discrete groups) whose quantum symmetries are much richer than their classical symmetry groups, and describes the associated quantum symmetry groups. The last lecture shows how functional analytic and geometric ideas can be used to detect and to quantify entanglement in high dimensions.

 The book will allow graduate students and young researchers to gain a better understanding of free probability, the theory of compact quantum groups, and applications of the theory of Banach spaces to quantum information. The latter applications will also be of interest to theoretical and mathematical physicists working in quantum theory.


Freeness Compact Quantum Group De Finetti Theorem Quantum Symmetry Group Quantum Isometry Group Entanglement Multipartite states

Authors and affiliations

  1. 1.Institut Camille JordanUniversité Claude Bernard Lyon 1LyonFrance
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  3. 3.Fachrichtung MathematikSaarland UniversitySaarbrückenGermany

Editors and affiliations

  • Uwe Franz
    • 1
  1. 1.Laboratoire de Mathématiques de BesançonUniversity Bourgogne Franche-ComtéBesançonFrance

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