Open Quantum Systems: A Mathematical Perspective PDF
Preview Open Quantum Systems: A Mathematical Perspective
Tutorials, Schools, and Workshops in the Mathematical Sciences Dorothea Bahns Anke Pohl Ingo Witt Editors Open Quantum Systems A Mathematical Perspective Tutorials, Schools, and Workshops in the Mathematical Sciences Thisserieswillserveasaresourceforthepublicationofresultsanddevelopments presentedatsummerorwinterschools,workshops,tutorials,andseminars.Written in an informal and accessible style, they present important and emerging topics in scientific research for PhD students and researchers. Filling a gap between traditional lecture notes, proceedings, and standard textbooks, the titles included inTSWMSpresentmaterialfromtheforefrontofresearch. Moreinformationaboutthisseriesathttp://www.springer.com/series/15641 Dorothea Bahns • Anke Pohl • Ingo Witt Editors Open Quantum Systems A Mathematical Perspective Editors DorotheaBahns AnkePohl MathematischesInstitut Department3–Mathematik UniversitätGöttingen UniversitätBremen Göttingen,Germany Bremen,Germany IngoWitt MathematischesInstitut UniversitätGöttingen Göttingen,Germany ISSN2522-0969 ISSN2522-0977 (electronic) Tutorials,Schools,andWorkshopsintheMathematicalSciences ISBN978-3-030-13045-9 ISBN978-3-030-13046-6 (eBook) https://doi.org/10.1007/978-3-030-13046-6 MathematicsSubjectClassification(2010):81S22,82C10,81P16,81P45,81P40,60J35,46L55,81Q93, 47D03,94A17 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Thetheoryofopenquantumsystems,thatis,quantumsystemsinteractingwithan environment, is a fascinating and multifaceted field. It is guided by the demands of models for the actual quantum physical experiments and their interpretational questions, and it draws heavily upon many different mathematical areas in order to merge and combine tools and methods to form a new theory. Of particular ∗ importanceare functionalanalysis, evolutionequations, semigroups,C -algebras, andprobabilitytheory. Themathematicaltheoryofopenquantumsystemsisratheryoungandstillina formativestage,withafewfirststreamlinesemerging.Anintensiveandrichdevel- opmentin future is to be expected.This book is a compilation of four articles by internationallyleadingexpertswhichprovideanintroductiontosomefundamental mathematicalaspectsrelevanttounderstandingopenquantumsystems. ThearticlebyAlexanderBeltonprovidesaself-containedintroductiontoMarko- vian semigroups on both a classical and quantum-mechanical level, eventually characterizingthegeneratorsofcertainquantumFellersemigroups. Dariusz Chrus´cin´ski discusses non-Markovian quantum dynamics, with both time-localandmemorykernelmasterequations.Itismostlybasedontheexample of n-level quantum systems. A recurrent theme in both the articles of Alexander BeltonandDariuszChrus´cin´skiistheconceptofcompletelypositivemaps,which iscentraltothetheoryofopenquantumsystems. NielsJacobandElianRhindstudythegeneratorsofFellersemigroupsbytaking advantage of techniques from microlocal analysis. Their article constitutes a first attempttoincludegeometricaspectsintotheframework.Itfocusesontheclassical theory, and it is an interesting open problem to generalize this formalism to the quantumcase. AlsoVojkanJaksic’sarticlefocusesontheclassicaltheory.Itisthefirstpartof a series of articles devoted to the notion of entropy. He introduces and carefully analyzes notions of classical entropy that allow for counterparts in quantum informationtheoryorquantumstatisticalmechanics.Theforthcomingpartsinthis serieswillbepublishedelsewhere. v vi Preface The originalstimulusforthisvolumeis a winterschoolondynamicalmethods in open quantum systems which was held at the Mathematical Institute of the University of Göttingen in November2016 and where AlexanderBelton, Dariusz Chrus´cin´ski,NielsJacob,andVojkanJaksicwerelecturers. Göttingen,Germany DorotheaBahns Bremen,Germany AnkePohl Göttingen,Germany IngoWitt December2018 Contents IntroductiontoClassicalandQuantumMarkovSemigroups .............. 1 AlexanderC.R.Belton IntroductiontoNon-MarkovianEvolutionofn-LevelQuantum Systems............................................................................ 55 DariuszChrus´cin´ski Aspects of Micro-LocalAnalysis and Geometry in the Study ofLévy-TypeGenerators........................................................ 77 NielsJacobandElianO.T.Rhind LecturesonEntropy.I:Information-TheoreticNotions..................... 141 VojkanJakšic´ vii Contributors Alexander C. R. Belton Department of Mathematics and Statistics, Lancaster University,Lancaster,UK Dariusz Chrus´cin´ski Institute of Physics, Faculty of Physics, Astronomy and InformaticsNicolausCopernicusUniversity,Torun,Poland NielsJacob SwanseaUniversity,Swansea,Wales,UK Vojkan Jakšic´ Department of Mathematics and Statistics, McGill University, Montreal,QC,Canada ElianO.T.Rhind SwanseaUniversity,Swansea,Wales,UK ix Introduction to Classical and Quantum Markov Semigroups AlexanderC.R.Belton Abstract Weprovideaself-containedandfast-pacedintroductiontothetheoriesof operatorsemigroups,Markovsemigroupsandquantumdynamicalsemigroups.The levelisappropriateforwell-motivatedgraduatestudentswhohaveabackgroundin analysisorprobabilitytheory,withthefocusonthecharacterisationofinfinitesimal generatorsforvariousclassesofsemigroups.ThetheoremsofHille–Yosida,Hille– Yosida–Ray,Lumer–PhillipsandGorini–Kossakowski–Sudarshan–Lindbladareall proved, with the necessary technical prerequisites explained in full. Exercises are providedthroughout. 1 Introduction These notesare an extensionof a series of lecturesgivenat the Winter Schoolon DynamicalMethodsin Open QuantumSystems held at Georg-August-Universität GöttingenduringNovember2016.Theselectureswereaimedatgraduatestudents with a background in analysis or probability theory. The aim has been to make thenotesself-containedbutbrief,sothattheyarewidelyaccessible.Exercisesare providedthroughout. WebeginwiththebasicsofthetheoryofoperatorsemigroupsonBanachspaces, anddevelopthisuptotheHille–YosidaandLumer–Phillipstheorems;theseprovide characterisationsforthegeneratorsofstronglycontinuoussemigroupsandstrongly continuous contraction semigroups, respectively. As those with a background in probabilitytheorymaynotbecomfortablewithallof thenecessarymaterialfrom functional analysis, this is covered rapidly at the start. The reader can find much moreonthesetopicsinDavies’sbook[9]. After these fundamentals, we recall some key ideas from probability theory. The correspondence between time-homogeneous Markov processes and Markov A.C.R.Belton((cid:2)) DepartmentofMathematicsandStatistics,LancasterUniversity,Lancaster,UK e-mail:[email protected] ©SpringerNatureSwitzerlandAG2019 1 D.Bahnsetal.(eds.),OpenQuantumSystems,Tutorials,Schools,andWorkshops intheMathematicalSciences,https://doi.org/10.1007/978-3-030-13046-6_1
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