Lecture Notes in Mathematics 1954 Editors: J.-M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris · · Philippe Biane Luc Bouten Fabio Cipriani · · Norio Konno Nicolas Privault Quanhua Xu Quantum Potential Theory Editors: Michael Schürmann Uwe Franz ABC Editors UweFranz MichaelSchürmann Départementdemathématiques InstitutfürMathematikundInformatik deBesancon Ernst-Moritz-Arndt-UniversitätGreifswald UniversitédeFranche-Comté Jahnstrasse15a 16,routedeGray 17487Greifswald 25030Besanconcedex Germany France [email protected] [email protected] and GraduateSchoolofInformationSciences TohokuUniversity 6-3-09Aramaki-Aza-Aoba,Aoba-ku Sendai980-8579 Japan ISBN:978-3-540-69364-2 e-ISBN:978-3-540-69365-9 DOI:10.1007/978-3-540-69365-9 LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2008932185 MathematicsSubjectClassification(2000):58B34,81R60,31C12,53C21 (cid:2)c 2008Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Coverdesign:SPiPublishingServices Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 springer.com Preface This volume contains the notes of lectures given at the School “Quantum Potential Theory: Structure and Applications to Physics”. This school was held at the Alfried Krupp Wissenschaftskolleg in Greifswald from February 26 to March 9, 2007. We thank the lecturers for the hard work they ac- complishedinpreparingandgivingtheselectures andinwritingthese notes. Theirlecturesgiveanintroductiontocurrentresearchintheirdomains,which is essentially self-contained and should be accessible to Ph.D. students. We hope that this volume will help to bring together researchers from the areas of classical and quantum probability, functional analysis and operator alge- bras, and theoretical and mathematical physics, and contribute in this way to developing further the subject of quantum potential theory. We are greatly indebted to the Alfried Krupp von Bohlen und Halbach- Stiftung for the financial support, without which the school would not have beenpossible.We arealsoverythankfulfor the supportby the Universityof Greifswaldandthe UniversityofFranche-Comt´e.Oneofthe organisers(UF) wassupportedbyaMarieCurieOutgoingInternationalFellowshipoftheEU (Contract Q-MALL MOIF-CT-2006-022137). Special thanks go to Melanie Hinz who helped with the preparation and organisationof the school and who took care of all of the logistics. Finally, we would like to thank all the students for coming to Greifswald and helping to make the school a success. Sendai and Greifswald, Uwe Franz June 2008 Michael Schu¨rmann v Contents Introduction.................................................. 1 Potential Theory in Classical Probability ..................... 3 Nicolas Privault 1 Introduction.......................................... 3 2 Analytic Potential Theory.............................. 4 3 Markov Processes ..................................... 24 4 Stochastic Calculus.................................... 31 5 Probabilistic Interpretations ............................ 46 References ................................................. 58 Introduction to Random Walks on Noncommutative Spaces................................... 61 Philippe Biane 1 Introduction.......................................... 61 2 Noncommutative Spaces and Random Variables........... 62 3 Quantum Bernoulli Random Walks ...................... 68 4 Bialgebras and Group Algebras ......................... 73 5 Random Walk on the Dual of SU(2)..................... 76 6 Random Walks on Duals of Compact Groups ............. 80 7 The Case of SU(n) .................................... 82 8 Choquet-Deny Theorem for Duals of Compact Groups ..... 87 9 The Martin Compactification of the Dual of SU(2) ........ 90 10 Central Limit Theorems for the Bernoulli Random Walk ... 94 11 The Heisenberg Group and the Noncommutative Brownian Motion ..................................... 100 12 Dilations for Noncompact Groups ....................... 106 13 Pitman’s Theorem and the Quantum Group SU (2) ....... 110 q References ................................................. 114 vii viii Contents Interactions between Quantum Probability and Operator Space Theory................................................. 117 Quanhua Xu 1 Introduction.......................................... 117 2 Completely Positive Maps .............................. 119 3 Concrete Operator Spaces and Completely Bounded Maps........................................ 120 4 Ruan’s Theorem: Abstract Operator Spaces .............. 126 5 Complex Interpolation and Operator Hilbert Spaces ....... 130 6 Vector-valued Noncommutative L -spaces ................ 132 p 7 Noncommutative Khintchine Type Inequalities............ 137 8 Embedding of OH into Noncommutative L .............. 156 1 References ................................................. 158 Dirichlet Forms on Noncommutative Spaces .................. 161 Fabio Cipriani 1 Introduction.......................................... 161 2 Dirichlet Forms on C∗-algebras and KMS-symmetric Semigroups........................................... 168 3 Dirichlet Forms in Quantum Statistical Mechanics......... 218 4 Dirichlet Forms and Differential Calculus on C∗-algebras ... 224 5 Noncommutative Potential Theory and Riemannian Geometry ............................................ 245 6 Dirichlet Forms and Noncommutative Geometry .......... 259 7 Appendix ............................................ 265 8 List of Examples ...................................... 270 References ................................................. 272 Applications of Quantum Stochastic Processes in Quantum Optics ........................................................ 277 Luc Bouten 1 Quantum Probability .................................. 277 2 Conditional Expectations............................... 286 3 Quantum Stochastic Calculus........................... 290 4 Quantum Filtering .................................... 298 References ................................................. 306 Quantum Walks .............................................. 309 Norio Konno Part I: Discrete-Time Quantum Walks 1 Limit Theorems ...................................... 311 2 Disordered Case ...................................... 350 3 Reversible Cellular Automata .......................... 357 4 Quantum Cellular Automata........................... 368 5 Cycle ............................................... 377 6 Absorption Problems.................................. 387 Contents ix Part II: Continuous-Time Quantum Walks 7 One-Dimensional Lattice .............................. 401 8 Tree ................................................ 410 9 Ultrametric Space .................................... 420 10 Cycle ............................................... 432 References ................................................. 441 Index......................................................... 453 Contributors Philippe Biane Institut d’´electronique et d’informatique Gaspard-Monge, 77454 Marne-la- Vall´ee Cedex 2, France [email protected] Luc Bouten Caltech Physical Measurement and Control, MC 266-33, 1200 E California Blvd, Pasadena, CA 91125,USA [email protected] Fabio Cipriani Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32 I-20133 Milano, Italy [email protected] Norio Konno Department of Applied Mathematics, Yokohama National University, 79-5 Tokiwadai, Yokohama 240-8501,Japan [email protected] Nicolas Privault Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong [email protected] Quanhua Xu UFRSciencesettechniques,Universit´edeFranche-Comt´e,16,routedeGray, F-25 030 Besanc¸on Cedex, France [email protected] xi Introduction The term potential theory comes from 19th century physics, where the fun- damental forces like gravityor electrostatic forces were described as the gra- dients of potentials, i.e. functions which satisfy the Laplace equation. Hence potentialtheorywasthestudyofsolutionsoftheLaplaceequation.Nowadays thefundamentalforcesinphysicsaredescribedbysystemsofnon-linearpar- tial differential equations such as the Einstein equations and the Yang-Mills equations,andthe Laplace equationarisesonly as a limiting case.Neverthe- less,theLaplaceequationisstillusedinapplicationsinmanyareasofphysics and engineering like heat conduction and electrostatics. And the term “po- tential theory” has survivedas a convenientlabel for the theory of functions satisfying the Laplace equation, i.e. so-called harmonic functions. In the 20th century, with the development of probability and stochastic processes,itwas discoveredthatpotential theoryis intimately relatedto the theory of Markov processes, in particular diffusion processes and Brownian motion.Thedistributionsofthese processesevolveaccordingtoaheatequa- tion,and invariantdistributions satisfy a Laplace-typeequation.Conversely, theseprocessescanbeusedtoexpresssolutionsof,e.g.,theLaplaceequation. Formoredetails seeNicolasPrivault’slecture “PotentialTheoryinClassical Probability” in this volume. The notions ofquantumstochasticprocessesandquantumMarkovproce- sses were introduced in the 1970’s and allow to describe open quantum systems in close analogy to classical probability and classical Markov proce- sses. Roughly speaking, one can now recognize two different trends in the subsequent development of the theory of quantum Markov processes. The firstisguidedbyphysicalapplications,studiesconcretephysicallymotivated models, and develops tools for filtering noisy quantum signals or controling noisy quantum systems. The second aims to develop a mathematical theory, by generalizing or extending key results of the theory of Markov processes to the quantum (or noncommutative) case, and by looking for analogues of important tools that greatly influenced the development of classical poten- tialtheory,like stochastic calculus,Dirichletforms,orboundaries ofrandom U.Franz,M.Schu¨rmann(eds.)Quantum Potential Theory. 1 LectureNotesinMathematics 1954. (cid:2)c Springer-VerlagBerlinHeidelberg2008 2 Introduction walks. In our school the first direction was represented by Luc Bouten’s lec- ture “Applications of Controlled Quantum Processes in Quantum Optics”, the second by Philippe Biane’s lecture “Introduction to Random Walks on Noncommutative Spaces” and by Fabio Cipriani’s lecture “Noncommutative Dirichlet Forms”, see also the corresponding chapters of this book. Besides providing important background material on operator algebras and noncommutative analogues of function spaces used in other lectures, Quanhua Xu’s lecture on “Interactions between Quantum Probability and Operator Space Theory” shows how quantum probability can be applied to modernfunctionalanalysis.Forexample,acleverchoiceofsequencesofquan- tum random variables plays an essential role in establishing key results like noncommutative Khintchine type inequalities. Centralquestionsfromprobabilisticpotentialtheorylikethecomputation ofhitting times andthestudy ofthe asymptoticbehaviourofa walkarealso the main topic in Norio Konno’s lecture on “Quantum Walks”. These quan- tum walks are not quantum Markov processes in the sense of the lectures by Biane, Bouten, and Cipriani, but another type of quantum analogue of random walks and Markov chains, and many of the classical potential the- oretical methods have interesting analogues adapted to this case. By giving an introduction and survey of this quickly developing field this lecture was an enrichment of the school and nicely complements the other chapters. The goalofthe School“QuantumPotentialTheory:Structure andAppli- cationstoPhysics”andtheselecturenotesistwo-fold.Firstofallwewantto provideanintroductiontotherapidlydevelopingtheoryofquantumMarkov semigroups and quantum Markov processes with its manifold aspects rang- ing from functional analysis and probability theory to quantum physics. We hope that we have succeeded in preparing a monograph that is accessible to graduate students in mathematics and physics. But furthermore we also hope that this book will catch the interest of experienced mathematicians and physicists working in this field or related fields, in order to stimulate more communication between researchers working on “pure” and “applied” aspects. We believe that a strong collaboration between these communities willbetoeverybody’sbenefit.Keepinginmindthephysicalapplicationswill help to sharpen the theoreticians’ eye for the relevant questions and prop- erties, and new powerful mathematical tools will allow to get a better and deeper understanding of concrete physical systems.