Quantum Symmetries: Metabief, France 2014

126 Pages·2017·2.401 MB·English

by  Guillaume Aubrun, Adam Skalski, Roland Speicher

Lecture Notes in Mathematics 2189
Guillaume Aubrun · Adam Skalski
Roland Speicher
Quantum
Symmetries
Metabief, France 2014
Uwe Franz Editor
Lecture Notes in Mathematics 2189
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryBoard:
MichelBrion,Grenoble
CamilloDeLellis,Zurich
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DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
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Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Guillaume Aubrun (cid:129) Adam Skalski (cid:129)
Roland Speicher
Quantum Symmetries
Metabief, France 2014
Uwe Franz
Editor
123
Authors
GuillaumeAubrun AdamSkalski
InstitutCamilleJordan InstituteofMathematics
UniversitéClaudeBernardLyon1 PolishAcademyofSciences
Lyon,France Warsaw,Poland
RolandSpeicher
FachrichtungMathematik
SaarlandUniversity
SaarbruRcken,Germany
Editor
UweFranz
LaboratoiredeMathématiquesdeBesançon
UniversityBourgogneFranche-Comté
Besançon,France
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-63205-6 ISBN978-3-319-63206-3 (eBook)
DOI10.1007/978-3-319-63206-3
LibraryofCongressControlNumber:2017948029
MathematicsSubjectClassification(2010):46L54,81P45,46L65,81P40,46L89,16T30,60B15,81T75,
16T05
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Preface
ThismonographarosefromlecturesdeliveredbytheauthorsattheWinterSchool
“OperatorSpaces,NoncommutativeProbabilityandQuantumGroups”,December
1–12, 2014, held at the Azuréva Métabief in the Jura Mountains in France. It
containsthreeoftheschool’ssixlectures:
(cid:129) GuillaumeAubrun:Quantumentanglementinhighdimensions,
(cid:129) BenoîtCollins:Weingartencalculusandapplications,
(cid:129) Gilles Pisier: Grothendieckinequalities,tensor productsof operatorspacesand
relatedtopics,
(cid:129) Zhong-Jin Ruan: Operator spaces and their applications to (quantum) group
algebras,
(cid:129) AdamSkalski:Quantumsymmetrygroupsandrelatedtopics,
(cid:129) RolandSpeicher:Freeprobabilityandnon-commutativesymmetries.
Theselecturesprovideanintroductiontocurrentresearchtopicsin functionaland
harmonicanalysisandtheirapplicationtothetheoryofquantuminformation.
Acknowledgements We would like to thank the Region of Franche-Comté for
its financial support and the Hotel Azuréva in Métabief for its hospitality. We
also express our gratitude to the lecturers who invested a lot of time and work in
preparinganddeliveringtheircourses.And,finally,wethankthestudentsandyoung
researcherswhocontributedgreatlytomakingourschoolasuccess.
Besançon,France UweFranz
March2017
v
TheParticipantsoftheFirstWeek
TheParticipantsoftheSecondWeek
Contents
1 Introduction .................................................................. 1
UweFranz
2 FreeProbabilityandNon-commutativeSymmetries .................... 5
RolandSpeicher
3 QuantumSymmetryGroupsandRelatedTopics ........................ 37
AdamSkalski
4 QuantumEntanglementinHighDimensions............................. 83
GuillaumeAubrun
Index............................................................................... 115
ix
Chapter 1
Introduction
UweFranz
Symmetries play a deep role in the study of our universe and its laws. Modern
philosophers suggest that the elementary entities that make up our world are not
particles or fields, but rather symmetries [Ku]. For mathematiciansand physicists
the notion of symmetry is intimately related to group theory. An ornament or a
crystalhasahighdegreeofsymmetryifitinvariantunderalargesymmetrygroup
[Wey],i.e.,underalargegroupofreflections,rotations,andtranslations.
Inthe1980sphysicistsandmathematiciansstartedtoexplore“quantumgroups”
asageneralisationofgroupsandspeculatedthatthesewouldbeusefultodescribe
more general notionsof symmetry at the level of quantumphysics [Dr, Wo]. The
first exampleswhere obtainedas continuousdeformationsof (simple)Lie groups,
butlaternewfamiliesofexamplesaroseasquantumsymmetrygroupsofalgebras,
graphs,spectraltriples,etc.,cf.[Wa1,Wa2,BBC,GB].Itfurthermorehasturnedout
thattherepresentationtheoryofthesenewquantumgroupsandtherelatedharmonic
analysishavemanycloseconnectionstononcommutativeprobability,inparticular
tofreeprobability.
This book originated from the Winter School “Operator Spaces, Noncommu-
tative Probability and Quantum Groups” in Métabief in December 2014, where
we explored applications of functional analysis to quantum group theory and to
noncommutativeprobability.
The first lecture in this volume deals with the interplay between symmetry
and independence in noncommutative probability. De Finetti’s famous theorem
characterizessequencesofexchangeablerandomvariablesasmixturesofindepen-
dent and identically distributed random variables, see [Ka]. In Theorem 2.3.8 in
RolandSpeicher’slectureon“FreeProbabilityandNon-commutativeSymmetries”
U.Franz((cid:2))
LaboratoiredeMathématiquesdeBesançon,UniversityBourgogneFranche-Comté,Besançon,
France
e-mail:[email protected]
©SpringerInternationalPublishingAG2017 1
G.Aubrunetal.,QuantumSymmetries,LectureNotesinMathematics2189,
DOI10.1007/978-3-319-63206-3_1