Quantum Independent Increment Processes II_ Structure of Quantum Levy Processes, Classical Prob

Lecture Notes in Mathematics Editors: J.-M Morel, Cachan F Takens, Groningen B Teissier, Paris 1866 www.pdfgrip.com Ole E Barndorff-Nielsen · Uwe Franz · Rolf Gohm Burkhard Kümmerer · Steen Thorbjørnsen Quantum Independent Increment Processes II Structure of Quantum Lévy Processes, Classical Probability, and Physics Editors: Michael Schüermann Uwe Franz ABC www.pdfgrip.com Editors and Authors Ole E Barndorff-Nielsen Department of Mathematical Sciences University of Aarhus Ny Munkegade, Bldg 350 8000 Aarhus Denmark e-mail: oebn@imf.au.dk Burkhard Kümmerer Fachbereich Mathematik Technische Universität Darmstadt Schlossgartenstr 64289 Darmstadt Germany e-mail: kuemmerer@mathematik tu-darmstadt.de Michael Schuermann Rolf Gohm Uwe Franz Institut für Mathematik und Informatik Universität Greifswald Friedrich-Ludwig-Jahn-Str 15a 17487 Greifswald Germany e-mail: schurman@uni-greifswald.de gohm@uni-greifswald.de franz@uni-greifswald.de Steen Thorbjørnsen Department of Mathematics and Computer Science University of Southern Denmark Campusvej 55 5230 Odense Denmark e-mail: steenth@imada.sdu.dk Library of Congress Control Number: 2005934035 Mathematics Subject Classification (2000): 60G51, 81S25, 46L60, 58B32, 47A20, 16W30 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-24407-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-24407-3 Springer Berlin Heidelberg New York DOI 10.1007/11376637 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors and Techbooks using a Springer LATEX package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11376637 41/TechBooks www.pdfgrip.com 543210 Preface This volume is the second of two volumes containing the lectures given at the School “Quantum Independent Increment Processes: Structure and Applications to Physics” This school was held at the Alfried Krupp Wissenschaftskolleg in Greifswald during the period March 9–22, 2003 We thank the lecturers for all the hard work they accomplished Their lectures give an introduction to current research in their domains that is accessible to Ph D students We hope that the two volumes will help to bring researchers from the areas of classical and quantum probability, operator algebras and mathematical physics together and contribute to developing the subject of quantum independent increment processes We are greatly indebted to the Volkswagen Foundation for their financial support, without which the school would not have been possible We also acknowledge the support by the European Community for the Research Training Network “QP-Applications: Quantum Probability with Applications to Physics, Information Theory and Biology” under contract HPRN-CT-200200279 Special thanks go to Mrs Zeidler who helped with the preparation and organisation of 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 Neuherberg and Greifswald, August 2005 Uwe Franz Michael Schă urmann www.pdfgrip.com www.pdfgrip.com Contents Random Walks on Finite Quantum Groups Uwe Franz, Rolf Gohm Markov Chains and Random Walks in Classical Probability Quantum Markov Chains Random Walks on Comodule Algebras Random Walks on Finite Quantum Groups Spatial Implementation Classical Versions Asymptotic Behavior A Finite Quantum Groups B The Eight-Dimensional Kac-Paljutkin Quantum Group References 11 12 18 22 24 26 30 Classical and Free Infinite Divisibility and L´ evy Processes Ole E Barndorff-Nielsen, Steen Thorbjørnsen 33 Introduction 34 Classical Infinite Divisibility and L´evy Processes 35 Upsilon Mappings 48 Free Infinite Divisibility and L´evy Processes 92 Connections between Free and Classical Infinite Divisibility 113 Free Stochastic Integration 123 A Unbounded Operators Affiliated with a W ∗ -Probability Space 150 References 155 L´ evy Processes on Quantum Groups and Dual Groups Uwe Franz 161 www.pdfgrip.com VIII Contents L´evy Processes on Quantum Groups 163 L´evy Processes and Dilations of Completely Positive Semigroups 184 The Five Universal Independences 198 L´evy Processes on Dual Groups 229 References 254 Quantum Markov Processes and Applications in Physics Burkhard Kă ummerer 259 Quantum Mechanics 262 Unified Description of Classical and Quantum Systems 265 Towards Markov Processes 268 Scattering for Markov Processes 281 Markov Processes in the Physics Literature 294 An Example on M2 297 The Micro-Maser as a Quantum Markov Process 302 Completely Positive Operators 308 Semigroups of Completely Positive Operators and Lindblad Generators 312 10 Repeated Measurement and its Ergodic Theory 315 References 328 Index 331 www.pdfgrip.com Contents of Volume I L´ evy Processes in Euclidean Spaces and Groups David Applebaum Introduction Lecture 1: Infinite Divisibility and L´evy Processes in Euclidean Space L´evy Processes Lecture 2: Semigroups Induced by L´evy Processes Analytic Diversions Generators of L´evy Processes Lp -Markov Semigroups and L´evy Processes Lecture 3: Analysis of Jumps Lecture 4: Stochastic Integration 10 Lecture 5: L´evy Processes in Groups 11 Lecture 6: Two L´evy Paths to Quantum Stochastics References 15 25 29 33 38 42 55 69 84 95 Locally compact quantum groups Johan Kustermans 99 Elementary C*-algebra theory 102 Locally compact quantum groups in the C*-algebra setting 112 Compact quantum groups 115 Weight theory on von Neumann algebras 129 The definition of a locally compact quantum group 144 Examples of locally compact quantum groups 157 Appendix : several concepts 172 References 176 Quantum Stochastic Analysis – an Introduction J Martin Lindsay 181 Spaces and Operators 183 QS Processes 214 QS Integrals 221 www.pdfgrip.com X Contents QS Differential Equations 238 QS Cocycles 243 QS Dilation 253 References 264 Dilations, Cocycles and Product Systems B V Rajarama Bhat 273 Dilation theory basics 273 E0 -semigroups and product systems 277 Domination and minimality 282 Product systems: Recent developments 286 References 290 Index 293 www.pdfgrip.com 326 Burkhard Kă ummerer n Yn := j=1 Vj j is a martingale From E(Vj2 ) ≤ · x we infer E(Yn2 ) ≤ · x · π6 , hence (Yn )n≥1 is uniformly bounded in L1 (Ω, Pϕ ) Thus, by the martingale convergence theorem (cf [Dur]), n Vj =: Y∞ j lim n→∞ j=1 exists Pϕ –almost surely Applying Kronecker’s Lemma (cf [Dur]), it follows that N N −1 Vj j=0 −→ Pϕ –almost surely, N →∞ i.e., N N −1 Θj+1 (x) − Θj (T x) −→ N →∞ j=0 Pϕ –almost surely, hence N N −1 Θj (x) − Θj (T x) j=0 −→ N →∞ Pϕ –almost surely, since the last sum differs from the foregoing only by two summands which can be neglected when N becomes large Applying T it follows that N N −1 Θj (T x) − Θj (T x) −→ N →∞ j=0 Pϕ –almost surely, and by adding this to the foregoing expression we obtain N N −1 Θj (x) − Θj (T x) j=0 −→ N →∞ Pϕ –almost surely By the same argument we see N N −1 Θj (x) − Θj (T l x) j=0 −→ N →∞ Pϕ –almost surely for all l ∈ N and averaging this over the first m values of l yields www.pdfgrip.com Quantum Markov Processes and Applications in Physics N N −1 Θj (x) − j=0 m 327 m−1 −→ Θj (T l x) N →∞ l=0 Pϕ –almost surely for m ∈ N We may exchange the limits N → ∞ and m → ∞ and finally obtain N N −1 Θj (x) − Θj (P x) j=0 −→ N →∞ Pϕ –almost surely (∗∗) Step 2: From the above key observation (∗) we obtain E(Θn+1 (P x)|Σn ) = Θn (T P x) = Θn (P x) , hence the process (Θn (P x))n≥0 , too, is a uniformly bounded martingale which x Pϕ –almost surely on Ω By (∗∗) the converges to a random variable Θ∞ averages of the difference (Θj (x) − Θj (P x))j≥0 converge to zero, hence N →∞ N N −1 x Θj (x) = Θ∞ lim Pϕ – almost surely on Ω j=0 This holds for all x ∈ A , hence the averages N N −1 Θj j=0 converge to some random variable Θ∞ with values in the state space of A Pϕ –almost surely Finally, since P T x = T x for x ∈ A , we obtain Θ∞ (T x) = limn→∞ Θn (P T x) = limn→∞ Θn (P x) = Θ∞ (x) , hence Θ∞ takes values in the stationary states If a quantum trajectory starts in a pure state ϕ it will clearly stay in the pure states for all times However, our computer simulations showed that even if initially starting with a mixed state there was a tendency for the state to ”purify” along a trajectory There is an obvious exception: If T is decomposed into a convex combination of automorphisms, i.e., if the operators are multiples of unitaries for all i ∈ Ω0 then a mixed state ϕ will never purify since all states along the trajectory will stay being unitarily equivalent to ϕ In a sense this is the only exception: For a state ψ on A = Mn we denote by ρψ the corresponding density matrix such that ψ(x) = tr(ρψ · x) where, as usual, tr denotes the trace on A = Mn www.pdfgrip.com 328 Burkhard Kă ummerer Denition 10.5 A quantum trajectory (n (ω))n≥0 purifies, if lim tr(ρ2Θn (ω) ) = n Theorem 10.6 [MaKă u] The quantum trajectories (n ())n0 , ω ∈ Ω , purify Pϕ –almost surely or there exists a projection p ∈ A = Mn with dim p ≥ 2, such that pa∗i p = λi p for all i ∈ Ω0 and λi ≥ Corollary 10.7 On A = M2 quantum trajectories purify Pϕ –almost surely or = λi ui for λi ∈ C and ui ∈ M2 unitary for all i ∈ Ω0 , i.e., T is decomposed into a convex combination of automorphisms References [AFL] [ApH] [BKS] [Car] [Dav1] [Dav2] [Dur] [Eva] [EvLe] [GKS] [Haa] [Hid] [Kra] [Kre] [Kă u1] [Kă u2] [Kă u3] L.Accardi, F Frigerio, J.T Lewis: Quantum stochastic processes Publ RIMS 18 (1982), 97 - 133 269 D Applebaum, R.L Hudson: Fermion Itˆ o’s formula and stochastic evolutions Commun Math Phys 96 (1984), 473 292 M Bo˙zejko, B Kă ummerer, R Speicher: q-Gaussian processes: non-commutative and classical aspects Commun Math Phys 185 (1997), 129 154 281, 287, 288 H J Carmichael: An Open Systems Approach to Quantum Optics Springer Verlag, Berlin 1993 314, 325 E.B Davies: Quantum Theory of Open Systems Academic Press, London 1976 324 E B Davies: One Parameter Semigroups Academic Press, London 1980 312 R Durett: Probability: Theory and Examples Duxbury Press, Belmont 1996 326 D E Evans: Completely positive quasi-free maps on the CAR algebra Commun Math Phys 70 (1979), 53-68 281 D Evans, J.T Lewis: Dilations of Irreversible Evolutions in Algebraic Quantum Theory Comm Dublin Inst Adv Stud Ser A 24, 1977 268, 278, 281 V Gorini, A Kossakowski, E.C.G Sudarshan: Completely positive dynamical semigroups of n-level systems, J Math Phys 17 (1976), 821 825 312 F Haag: Asymptotik von Quanten-Markov-Halbgruppen und QuantenMarkov-Prozessen, Dissertation, Darmstadt 2005 302, 307, 308 T Hida: Brownian motion Springer-Verlag, Berlin 1980 261, 280, 287, 288 K Kraus: General state changes in quantum theory Ann Phys 64 (1971), 311 - 335 317 U Krengel: Ergodic Theorems Walter de Gruyter, Berlin-New York 1985 325 B Kă ummerer: Examples of Markov dilations over the × -matrices In Quantum Probability and Applications I, Lecture Notes in Mathematics 1055, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo 1984, 228 - 244 297 B Kă ummerer: Markov dilations on W*-algebras Journ Funct Anal 63 (1985), 139 - 177 275, 278, 279, 301, 302 B Kă ummerer: Stationary processes in quantum probability Quantum Probability Communications XI World Scientific 2003, 273 - 304 262, 277, 287, 288 www.pdfgrip.com Quantum Markov Processes and Applications in Physics 329 B Kă ummerer: Quantum Markov processes In Coherent Evolution in Noisy Environments, A Buchleitner, K Hornberger (Eds.), Springer Lecture Notes in Physics 611 (2002), 139 - 198 262 [Kă uMa1] B Kă ummerer, H Maassen: The essentially commutative dilations of dynamical semigroups on Mn Commun Math Phys 109 (1987), - 22 293, 301, 313, 314 [Kă uMa2] B Kă ummerer, H Maassen: Elements of quantum probability In Quantum Probability Communications X, World Scientific 1998, 73 - 100 265, 287, 288 [Kă uMa3] B Kă ummerer, H Maassen: A scattering theory for Markov chains Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol 3, No (2000), 161 - 176 278, 281, 286, 290, 291, 292, 302 [Kă uMa4] B Kă ummerer, H Maassen: An ergodic theorem for quantum counting processes J Phys A: Math Gen 36 (2003), 2155 - 2161 314, 323 [Kă uMa5] B Kă ummerer, H Maassen: A pathwise ergodic theorem for quantum trajectories J Phys A: Math Gen 37 (2004) 11889-11896 314, 324 [Kă uNa] B Kă ummerer, R.J Nagel: Mean ergodic semigroups on W*-Algebras Acta Sci Math 41 (1979), 151-159 279, 281, 325 [Kă uS1] B Kă ummerer, W Schră oder: A new construction of unitary dilations: singular coupling to white noise In Quantum Probability and Applications II, (L Accardi, W von Waldenfels, eds.) Springer, Berlin 1985, 332347 (1985) 285 [Kă uS2] B Kă ummerer, W Schră oder: A Markov dilation of a non-quasifree Bloch evolution Comm Math Phys 90 (1983), 251-262 297 [LaPh] P.D Lax, R.S Phillips: Scattering Theory Academic Press, New York 1967 278, 286, 288, 290 [Lin] G Lindblad: On the generators of quantum dynamical semigroups, Commun Math Phys 48 (1976), 119 - 130 312 [LiMa] J.M Lindsay, H Maassen: Stochastic calculus for quantum Brownian motion of non-minimal variance In: Mark Kac seminar on probability and physics, Syllabus 1987–1992 CWI Syllabus 32 (1992), Amsterdam 292, 293 [MaKă u] H Maassen, B Kă ummerer: Purification of quantum trajectories, quantph/0505084, to appear in IMS Lecture Notes-Monograph Series 328 [Mol] B.R Mollow: Power spectrum of light scattered by two-level systems Phys Rev 188 (1969), 1969–1975 291 [JvN] John von Neumann: Mathematische Grundlagen der Quantenmechanik Springer, Berlin 1932, 1968 262 [Par] K.R Parthasarathy: An Introduction to Quantum Stochastic Calculus Birkhă auser Verlag, Basel 1992 274, 292, 315 [RoMa] P Robinson, H Maassen: Quantum stochastic calculus and the dynamical Stark effect Reports Math Phys 30 (1991), 185–203 291, 293, 294 [RS] M Reed, B Simon: Methods of Modern Mathematical Physics I: Functional Analysis Academic Press, New York 1972 263 [SSH] F Schuda, C.R Stroud, M Hercher: Observation of resonant Stark effect at optical frequencies Journ Phys B7 (1974), 198 291 [SzNF] B Sz.-Nagy, C Foias: Harmonic Analysis of Operators on Hilbert Space North Holland, Amsterdam 1970 278, 282, 288 [Tak1] M Takesaki: Conditional expectations in von Neumann algebras J Funct Anal (1971), 306 - 321 272, 288 [Tak2] M Takesaki: Theory of Operator Algebras I Springer, New York 1979 266, 267, 271, 272, 275, 309, 310, [VBWW] B.T.H Varcoe, S Battke, M Weidinger, H Walther: Preparing pure photon number states of the radiation field Nature 403 (2000), 743 - 746 302, 306 [Kă u4] www.pdfgrip.com 330 Burkhard Kă ummerer [WBKM] T Wellens, A Buchleitner and B Kă ummerer, H Maassen: Quantum state preparation via asymptotic completeness Phys Rev Letters 85 (2000), 3361 302, 306, 308 [Wel] Thomas Wellens: Entanglement and Control of Quantum States Dissertation, Mă unchen 2002 308 www.pdfgrip.com Index ∗ -Hopf algebra 24 ∗ -bialgebra 24 ∗ –homomorphism 266 ∗ -algebra 262 absolute continuity 76 additive process classical 35 free 111, 121 adjoint 262 algebra of observables 262 anti-monotone calculus 247 anti-monotone L´evy process 230 anti-monotone product 216 anti-monotonically independent 230 anti-symmetric independence 215 antipode 24, 163, 229 arrow see morphism associativity property 196 asymptotic completeness 286, 303 asymptotically complete 287 automorphism 197, 275 background driving L´evy process (BDLP) 44 bath 291 Bercovici-Pata bijection Λ 112, 114, 124, 137 algebraic properties 114 coneection between Λ and Υ 113 topological properties 116 bialgebra 24 involutive 24, 162 binary product 201 Birkhoff ergodic theorem 319 boolean calculus 246 boolean independence 230 boolean L´evy process 230 boolean product 216 Bose independence 162, 214 Brownian motion 258 C∗ -algebra 92 -probability space 93 C*-algebra 262, 264 canonical anticommutation relations 277 canonical commutation relations 17, 277 canonical realization of a stochastic process 273 categorical equivalence 201 categories isomorphic 201 category 196, 275 dual 198 full sub- 198 functor 201 monoidal 207 opposite 198 sub- 198 category of algebraic probability spaces 213 Cauchy transform 100, 152 Cayley transform 97 CCR flow 185 characteristic triplet www.pdfgrip.com 332 Index classical 35 free 103 coaction coboundary 166, 168 cocycle 166 codomain see target cogroup 229 comodule algebra comonoidal functor 208 complete monotonicity 46, 67 completely positive 305 completely positive operator 276 components of a natural transformation 200 compound Poisson process 169 comultiplication 24, 162, 229 concrete representation of completely positive operators 307, 317 conditional expectation 268, 276, 291, 313 conditional expectation of tensor type 268, 271, 291, 314 conditionally positive 165 contravariant functor 199 convolution of algebra homomorphisms 229 of linear maps 162 convolution semigroup of states 163 coproduct 204 of a bialgebra 24, 162 coproduct injection 205 cotensor functor 208 counit 24, 229 coupling to a shift covariant functor 199 cumulant transform classical 34 free 99 cumulants 101, 113 density matrix 260, 263, 265 density operator 263 diffusion term 310 dilation 191, 274, 275 dilation diagram 275 distribution 261, 266 of a quantum random variable 161 of a quantum stochastic process 161 domain see source drift 168 drift term 310 dual category 198 dual group 229 dual semigroup 229 in a tensor category 233 endomorphism 197 epi see epimorphism epimorphism 198 equilibrium distribution 257 equivalence 201 categorical 201 natural see natural isomorphism equivalence of quantum stochastic processes 161 ergodic theorem 319–321 expectation 259 expectation functional 263 expectation value 263 factorizable representation 179 Fermi independence 215 finite quantum group 25 flip 13, 26, 163, 231 free additive convolution 98, 100 Brownian motion 121 cumulant transform 99 cumulants 101 independence 94, 97, 151 and weak convergence 95 infinite divisibility 102 free independence 216 free L´evy process 230 free product of ∗ -algebras 205, 207 of states 216 freely independent 230 full subcategory 198 function 197 total 197 functor 199 comonoidal 208 contravariant 199 covariant 199 identity 199 monoidal 208 www.pdfgrip.com Index functor category 201 functor of white noise 277, 285, 286 fundamental operator 12, 26 Gaussian generator 168 Gaussian process 277 generalized inverse Gaussian distribution 45 generating pair classical 34 free 103 and weak convergence 116 generator Poisson 169 quadratic or Gaussian 168 generator of a L´evy process 165 GIG see generalized inverse Gaussian distribution Gnedenko 116 GNS representation 134 Goldie-Steutel-Bondesson class 43, 46 H-algebra 229 Haar measure 183 Haar state 25, 28 Heisenberg picture 265 hom-set 197 Hopf algebra 163 involutive 24 HP-cocycle 171, 185 identical natural transformation 201 identity functor 199 identity morphism 196 identity property 196 inclusion 199 increment property 162 independence 283 anti-monotone 230 boolean 230 Bose or tensor 162, 214 Fermi or anti-symmetric 215 free 216, 230 monotone 230 of morphisms 211 tensor 230 independent stochastically 209 infinite divisibility 34 333 classical 34 characterization of classes of laws in terms of L´evy measure 46 classes of laws 42 free 91, 102 classes of laws 105, 115 of matrices 90 initial distribution initial object 205 initial state injection coproduct 205 inner automorphism 295 integral representation see stochastic integration inverse left 198 right 198 inverse Gaussian distribution 45 inverse morphism 197 invertible morphism see isomorphism involutive bialgebra 24, 162 involutive Hopf algebra 24, 163 irreducibility axiom 262 isomomorphism natural 200 isomorphic 197 isomorphic categories 201 isomorphism 197 joint distribution of a quantum stochastic process jump operator 310 161 Laplace like transform 77 left inverse 198 leg notation 13, 26 L´evy copulas 89 measure 35 process classical 34, 35 connection between classical and free 120, 125 free 91, 110, 122 on a dual semigroup 230 on a Hopf ∗ -algebra 163 on a Lie algebra 177 on an involutive bialgebra 162 www.pdfgrip.com 334 Index L´evy-Itˆ o decomposition classical 40, 41 free 139, 143, 145 L´evy-Khintchine representation classical 34 free 102, 103 Lie algebra 177 Lindblad form of a generator 309 marginal distribution of a quantum stochastic process 161 Markov chain Markov process 3, 265, 270, 273, 292, 311 Markov property 257 mean ergodic 321 measure topology 96, 151 measurement 259, 261 micro-maser 257, 313, 318 minimal concrete representation of completely positive operators 308 minimal dilation 192 minimal Stinespring representation 307 Mittag-Leffler distribution 73 function 72, 80 mixed state 260 module property 268 Mă obius transform 101 monic see monomorphism monoidal category 207 monoidal functor 208 monomorphism 198 monotone calculus 249 monotone L´evy process 230 monotone product 216 monotonically independent 230 morphism 196, 275 inverse 197 left 198 right 198 invertible see isomorphism morphism of functors see natural transformation multiplicative unitary 13, 26 n–positive 305 natural equivalence see natural isomomorphism natural isomorphism 200 natural transformation 200 identical 201 non-commutative analogue of the algebra of coefficients of the unitary group 182 non-commutative probability space 263 noncommutative probability 92 normal state 265 nuclear magnetic resonance 295 object 196 initial 205 terminal 204 observable 259 open system 291 operator algebra 257, 264 operator process 162 operator theory 92 opposite category 198 Ornstein-Uhlenbeck process OU process 128 P-representation 293 partial trace 292 pendagon axiom 207 perfect measurement 316 phase space methods 293 Pick functions 102 Poisson distribution classical 113 free 113 intensity measure classical 41 free 130 random measure classical 40 free 129, 130, 135 Poisson generator 169 positive definite 279 probability space 263, 276 product 201, 203 binary 201 product projection 204 projection www.pdfgrip.com 277 Index product 204 pufification of quantum trajectories 324 pure state 260 q-commutation relations 277 quadratic generator 168 quantum Az´ema martingale 181 quantum dynamical semigroup 191 quantum group finite 25 quantum Markov chain quantum measurement 312 quantum mechanics 259 quantum probability space 161, 263 quantum random variable 161 quantum regression theorem 270 quantum stochastic calculus 311 quantum stochastic process 161 quantum trajectory 311, 320, 321 quantum trajectory, purification 324 random variable 266, 276 random walk 1, 4, on a finite group random walk on a comodule algebra random walk on a finite quantum group 11 real-valued random variable 265 reciprocal inverse Gaussian distribution 45 reduced time evolution 292 reduction of an independence 218 repeated quantum measurement 317 representation theorem for L´evy processes on involutive bialgebras 169 retraction see left inverse right inverse 198 Schoenberg correspondence 240 Schră odinger picture 265 Schă urmann triple 166, 243 Schwarz inequality for maps 306 section see right inverse selfadjoint operator affiliated with W ∗ -algebra 93, 149, 150 spectral distribution 93, 94, 150 335 selfdecomposability classical 42–45 and Thorin class 65, 69 integral representation 44 free 105, 109 integral representation 125, 128 selfdecomposable laws classical 42–46, 89 free 105, 115 semi-circle distribution 95, 107, 112 source 196 spectral measure 261 spectral theorem 260 spectrum 260, 312 spin- 12 -particle 257, 259, 295 stable distributions classical 42–44, 46 free 105, 107 state 92, 259, 263, 264 normal 93 tracial 93 stationarity of increments 163 stationary Markov process 273 stationary state 319 stationary stochastic process 257, 267, 276 Stieltjes transform see Cauchy transform Stinespring representation 307, 317 stochastic differential equation 311 stochastic integration classical 36, 38 existence 38 free 122, 135, 136 connection 125 stochastic matrix stochastic process 266 stochastically independent 209 strong operator topology 264 subcategory 198 full 198 surjective Schă urmann triple 166 Sweedlers notation 162 target 196 tempered stable 42, 43, 46 tensor algebra 175 tensor category 207 with inclusions 211 www.pdfgrip.com 336 Index with projections 210 tensor functor 208 tensor independence 162, 214, 230 tensor independent tensor L´evy process 230 tensor product 207 terminal object 204 Thorin class connection to selfdecomposability 65, 69 general 42, 43, 46, 67, 69 positive 44, 62, 65 time translation 267, 276 total function 197 transformation identical natural 201 natural 200 transition matrix transition operator 7, 269, 270, 274, 276, 292, 295, 311 transition state triangle axiom 208 two–level system 259 unitary dilation 274, 276 unitization 207 universal enveloping algebra 178 unravellings of operators 311 Upsilon transformations 47, 86 Υ 54 Υ α 72, 79 Υ τ 87 Υ0 47 Υ0α 72, 74 absolute continuity 49, 76 algebraic properties 58, 82 connection Υ and Λ 113 connection to L(∗) and T (∗) 62 for matrix subordinators 89 generalized 86, 87 stochastic integral representation 85 Voiculescu transform 99, 100, 102, 106 von Neumann algebra 92, 149, 262, 264 W∗ -probability space 93, 149 W*-algebra 262 weak convergence 36, 37, 82, 100, 105 white noise 258, 292 Wigner 95 Wigner representation 293 www.pdfgrip.com Lecture Notes in Mathematics For information about earlier volumes please contact your bookseller or Springer LNM Online archive: springerlink.com Vol 1674: G Klaas, C R Leedham-Green, W Plesken, Linear Pro-p-Groups of Finite Width (1997) 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Lectures on Seiberg-Witten Invariants 1997 – Second Edition (2001) Vol 1638: P Vanhaecke, Integrable Systems in the realm of Algebraic Geometry 1996 – Second Edition (2001) Vol 1702: J Ma, J Yong, Forward-Backward Stochastic Differential Equations and their Applications 1999 – Corrected 3rd printing (2005) www.pdfgrip.com ... concept in classical probability is the notion of Markovianity In classical probability the class of Markov processes contains the class of processes with independent and stationary processes, ... · Rolf Gohm Burkhard Kümmerer · Steen Thorbjørnsen Quantum Independent Increment Processes II Structure of Quantum Lévy Processes, Classical Probability, and Physics Editors: Michael Schüermann... from the areas of classical and quantum probability, operator algebras and mathematical physics together and contribute to developing the subject of quantum independent increment processes We are