Alice and Bob meet Banach: the interface of asymptotic geometric analysis and quantum information theory
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2017]
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| Schriftenreihe: | Mathematical surveys and monographs
223 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xxi, 414 Seiten Illustration, Diagramme |
| ISBN: | 9781470434687 |
Internformat
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Datensatz im Suchindex
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| adam_text | Contents List of Tables xiii List of Figures xv Preface xix Part 1. Alice and Bob: Mathematical Aspects of Quantum Information Theory 1 Chapter 0. Notation and basic concepts 0.1. Asymptotic and nonasymptoticnotation 0.2. Euclidean and Hilbert spaces 0.3. Bra-ket notation 0.4. Tensor products 0.5. Complexification 0.6. Matrices vs. operators 0.7. Block matrices vs. operators onbipartite spaces 0.8. Operators vs. tensors 0.9. Operators vs. superoperators 0.10. States, classical and quantum 3 3 3 4 6 6 7 8 8 8 8 Chapter 1. Elementary convex analysis 1.1. Normed spaces and convex sets 1.1.1. Gauges 1.1.2. First examples: ^p-balls,simplices, polytopes, and convex hulls 1.1.3. Extreme points, faces 1.1.4. Polarity 1.1.5. Polarity and the facialstructure 1.1.6. Ellipsoids 1.2. Cones 1.2.1. Cone duality 1.2.2. Nondegenerate cones and facial structure 1.3. Majorization and Schatten norms 1.3.1. Majorization 1.3.2. Schatten norms 1.3.3. Von Neumann and Rényi entropies Notes and Remarks 11 11 11 12 13 15 17 18 18 19 21 22 22 23 27 29 Chapter 2. The mathematics of quantum information theory 2.1. On the geometry of the set of quantum states 2.1.1. Pure and mixed states 31 31 31 vii
viii CONTENTS 2.1.2. The Bloch ball D(C2) 2.1.3. Facial structure 2.1.4. Symmetries 2.2. States on multipartite Hilbert spaces 2.2.1. Partial trace 2.2.2. Schmidt decomposition 2.2.3. A fundamental dichotomy: Separability vs. entanglement 2.2.4. Some examples of bipartite states 2.2.5. Entanglement hierarchies 2.2.6. Partial transposition 2.2.7. PPT states 2.2.8. Local unitaries and symmetries of Sep 2.3. Superoperators and quantum channels 2.3.1. The Choi and Jamiołkowski isomorphisms 2.3.2. Positive and completely positive maps 2.3.3. Quantum channels and Stinespring representation 2.3.4. Some examples of channels 2.4. Cones of QIT 2.4.1. Cones of operators 2.4.2. Cones of superoperators 2.4.3. Symmetries of the PST cone 2.4.4. Entanglement witnesses 2.4.5. Proofs of Størmer’s theorem Notes and Remarks Chapter 3. Quantum mechanics for mathematicians 3.1. Simple-minded quantum mechanics 3.2. Finite vs. infinite dimension, projective spaces, and matrices 3.3. Composite systems and quantum marginals: Mixed states 3.4. The partial trace: Purification of mixed states 3.5. Unitary evolution and quantum operations: The completely positive maps 71 3.6. Other measurement schemes 3.7. Local operations 3.8. Spooky action at a distance Notes and Remarks 32 33 34 35 35 36 37 39 41 41 43 46 47 47 48 50 52 55 55 56 58 60 62 63 67 67 68 68 70 73 74 75 75 Part 2. Banach and His Spaces: Asymptotic Geometric Analysis Miscellany 77 Chapter 4. More convexity 4.1. Basic notions and operations 4.1.1. Distances between convex sets 4.1.2. Symmetrization 4.1.3. Zonotopes and zonoids 4.1.4.
Projective tensor product 4.2. John and Löwner ellipsoids 4.2.1. Definition and characterization 4.2.2. Convex bodies with enough symmetries 79 79 79 80 81 82 84 84 89
CONTENTS 4.2.3. Ellipsoids and tensor products 4.3. Classical inequalities for convexbodies 4.3.1. The Brunn-Minkowski inequality 4.3.2. log-concave measures 4.3.3. Mean width and the Urysohn inequality 4.3.4. The Santaló and the reverse Santaló inequalities 4.3.5. Symmetrization inequalities 4.3.6. Functional inequalities 4.4. Volume of central sections and theisotropic position Notes and Remarks ix 91 91 91 93 94 98 98 101 101 103 Chapter 5. Metric entropy and concentration of measure in classical spaces 5.1. Nets and packings 5.1.1. Definitions 5.1.2. Nets and packings on the Euclidean sphere 5.1.3. Nets and packings in the discrete cube 5.1.4. Metric entropy for convex bodies 5.1.5. Nets in Grassmann manifolds, orthogonal and unitary groups 5.2. Concentration of measure 5.2.1. A prime example: concentration on the sphere 5.2.2. Gaussian concentration 5.2.3. Concentration tricks and treats 5.2.4. Geometric and analytic methods. Classical examples 5.2.5. Some discrete settings 5.2.6. Deviation inequalities for sums of independent random variables Notes and Remarks 107 107 107 108 113 114 116 117 119 121 124 129 136 139 142 Chapter 6. Gaussian processes and random matrices 6.1. Gaussian processes 6.1.1. Key example and basic estimates 6.1.2. Comparison inequalities for Gaussian processes 6.1.3. Sudakov and dual Sudakov inequalities 6.1.4. Dudley’s inequality and the generic chaining 6.2. Random matrices 6.2.1. oo-Wasserstein distance 6.2.2. The Gaussian Unitary Ensemble (GUE) 6.2.3. Wishart matrices 6.2.4. Real RMT models and Chevet-Gordon inequalities 6.2.5. A quick
initiation to free probability Notes and Remarks 149 149 150 152 154 157 160 161 162 166 173 176 178 Chapter 7. Some tools from asymptotic geometric analysis 7.1. ¿-position, K-convexity and the MM*-estimate 7.1.1. ¿-norm and ¿-position 7.1.2. LC-convexity and the MM*-estimate 7.2. Sections of convex bodies 7.2.1. Dvoretzky’s theorem for Lipschitz functions 7.2.2. The Dvoretzky dimension 7.2.3. The Figiel֊Lindenstrauss-Milman inequality 181 181 181 182 186 186 189 193
X CONTENTS 7.2.4. The Dvoretzky dimension of standard spaces 7.2.5. Dvoretzky’s theorem forgeneral convex bodies 7.2.6. Related results 7.2.7. Constructivity Notes and Remarks Part 3. The Meeting: AGA and QIT 195 200 201 205 207 211 Chapter 8. Entanglement of pure states in high dimensions 8.1. Entangled subspaces: Qualitative approach 8.2. Entropies of entanglement and additivity questions 8.2.1. Quantifying entanglement for pure states 8.2.2. Channels as subspaces 8.2.3. Minimal output entropy and additivity problems 8.2.4. On the 1 — p norm of quantum channels 8.3. Concentration of Ep for p 1 and applications 8.3.1. Counterexamples to the multiplicativity problem 8.3.2. Almost randomizing channels 8.4. Concentration of von Neumann entropy and applications 8.4.1. The basic concentration argument 8.4.2. Entangled subspaces of small codimension 8.4.3. Extremely entangled subspaces 8.4.4. Counterexamples to the additivity problem 8.5. Entangled pure states in multipartite systems 8.5.1. Geometric measure of entanglement 8.5.2. The case of many qubits 8.5.3. Multipartite entanglement in real Hilbert spaces Notes and Remarks 213 213 215 215 216 216 217 218 218 220 222 222 224 224 228 229 229 230 231 232 Chapter 9. Geometry of the set of mixed states 9.1. Volume and mean width estimates 9.1.1. Symmetrization 9.1.2. The set of all quantum states 9.1.3. The set of separable states (the bipartitecase) 9.1.4. The set of block-positive matrices 9.1.5. The set of separable states (multipartitecase) 9.1.6. The set of PPT states 9.2. Distance estimates 9.2.1. The Gurvits-Barnum theorem
9.2.2. Robustness in the bipartite case 9.2.3. Distances involving the set of PPT states 9.2.4. Distance estimates in the multipartite case 9.3. The super-picture: Classes of maps 9.4. Approximation by polytopes 9.4.1. Approximating the set of all quantum states 9.4.2. Approximating the set of separable states 9.4.3. Exponentially many entanglement witnesses are necessary Notes and Remarks 235 236 236 236 238 240 242 244 245 246 247 248 249 250 252 252 256 258 260
CONTENTS xi Chapter 10. Random quantum states 10.1. Miscellaneous tools 10.1.1. Majorization inequalities 10.1.2. Spectra and norms of unitarily invariant random matrices 10.1.3. Gaussian approximation to induced states 10.1.4. Concentration for gauges of induced states 10.2. Separability of random states 10.2.1. Almost sure entanglement for low-dimensionalenvironments 10.2.2. The threshold theorem 10.3. Other thresholds 10.3.1. Entanglement of formation 10.3.2. Threshold for PPT Notes and Remarks 263 263 263 264 266 267 268 268 269 271 271 272 272 Chapter 11. Bell inequalities and the Grothendieck-Tsirelson inequality 11.1. Isometrically Euclidean subspaces via Clifford algebras 11.2. Local vs. quantum correlations 11.2.1. Correlation matrices 11.2.2. Bell correlation inequalities and the Grothendieck constant 11.3. Boxes and games 11.3.1. Bell inequalities as games 11.3.2. Boxes and the nonsignaling principle 11.3.3. Bell violations Notes and Remarks 275 275 276 277 280 283 284 285 289 294 Chapter 12. POVMs and the distillability problem 12.1. POVMs and zonoids 12.1.1. Quantum state discrimination 12.1.2. Zonotope associated to a POVM 12.1.3. Sparsification of POVMs 12.2. The distillability problem 12.2.1. State manipulation via LOCC channels 12.2.2. Distillable states 12.2.3. The case of two qubits 12.2.4. Some reformulations of distillability Notes and Remarks 299 299 299 300 300 301 301 302 302 304 305 Appendix A. Gaussian measures and Gaussian variables A.l. Gaussian random variables A.2. Gaussian vectors Notes and Remarks 307 307 308 309 Appendix B. B.l. The В.2. The
B.3. The B.4. The B.5. The Notes and Classical groups and manifolds unit sphere 5 ՜1 or Sc¿ projective space orthogonal and unitary groupsO(n), U(n) Grassmann manifolds Gr(fc,IR ),Gr(fc,C”) Lorentz group 0(1, n — 1) Remarks 311 311 312 312 314 318 319
CONTENTS xii Appendix C. Extreme maps between Lorentz cones and the S-lemma Notes and Remarks 321 324 Appendix D. Polarity and the Santaló point via duality of cones 325 Appendix E. Hints to exercises 329 Appendix F. Notation General notation Convex geometry Linear algebra Probability Geometry and asymptotic geometric analysis Quantum information theory 375 375 375 376 377 378 379 Bibliography Websites 381 408 Index 409
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| isbn | 9781470434687 |
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| spelling | Aubrun, Guillaume 1981- (DE-588)1141319640 aut Alice and Bob meet Banach the interface of asymptotic geometric analysis and quantum information theory Guillaume Aubrun ; Stanisław J. Szarek Providence, Rhode Island American Mathematical Society [2017] © 2017 xxi, 414 Seiten Illustration, Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs 223 Quanteninformatik (DE-588)4705961-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Geometrische Analysis (DE-588)4156708-0 gnd rswk-swf Quanteninformatik (DE-588)4705961-8 s Geometrische Analysis (DE-588)4156708-0 s Funktionalanalysis (DE-588)4018916-8 s DE-604 Szarek, Stanisław J. 1953- (DE-588)1141263165 aut Erscheint auch als Online-Ausgabe 978-1-4704-4172-2 Mathematical surveys and monographs 223 (DE-604)BV000018014 223 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029843966&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Aubrun, Guillaume 1981- Szarek, Stanisław J. 1953- Alice and Bob meet Banach the interface of asymptotic geometric analysis and quantum information theory Mathematical surveys and monographs Quanteninformatik (DE-588)4705961-8 gnd Funktionalanalysis (DE-588)4018916-8 gnd Geometrische Analysis (DE-588)4156708-0 gnd |
| subject_GND | (DE-588)4705961-8 (DE-588)4018916-8 (DE-588)4156708-0 |
| title | Alice and Bob meet Banach the interface of asymptotic geometric analysis and quantum information theory |
| title_auth | Alice and Bob meet Banach the interface of asymptotic geometric analysis and quantum information theory |
| title_exact_search | Alice and Bob meet Banach the interface of asymptotic geometric analysis and quantum information theory |
| title_full | Alice and Bob meet Banach the interface of asymptotic geometric analysis and quantum information theory Guillaume Aubrun ; Stanisław J. Szarek |
| title_fullStr | Alice and Bob meet Banach the interface of asymptotic geometric analysis and quantum information theory Guillaume Aubrun ; Stanisław J. Szarek |
| title_full_unstemmed | Alice and Bob meet Banach the interface of asymptotic geometric analysis and quantum information theory Guillaume Aubrun ; Stanisław J. Szarek |
| title_short | Alice and Bob meet Banach |
| title_sort | alice and bob meet banach the interface of asymptotic geometric analysis and quantum information theory |
| title_sub | the interface of asymptotic geometric analysis and quantum information theory |
| topic | Quanteninformatik (DE-588)4705961-8 gnd Funktionalanalysis (DE-588)4018916-8 gnd Geometrische Analysis (DE-588)4156708-0 gnd |
| topic_facet | Quanteninformatik Funktionalanalysis Geometrische Analysis |
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