Geometry of quantum states: an introduction to quantum entanglement
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2008
|
| Ausgabe: | repr. with corr., paperback ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 419 S. Ill., graph. Darst. |
| ISBN: | 9780521891400 9780521814515 |
Internformat
MARC
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| 100 | 1 | |a Bengtsson, Ingemar |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Geometry of quantum states |b an introduction to quantum entanglement |c Ingemar Bengtsson and Karol Życzkowski |
| 250 | |a repr. with corr., paperback ed. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2008 | |
| 300 | |a XII, 419 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Information theory | |
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| 700 | 1 | |a Życzkowski, Karol |e Verfasser |4 aut | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-017629034 | ||
Datensatz im Suchindex
| _version_ | 1804139228078014464 |
|---|---|
| adam_text | Contents
Preface page v
1
Convexity, colours and statistics
1
1.1
Convex sets
......................... 1
1.2
High-dimensional geometry
................ 7
1.3
Colour theory
....................... 12
1.4
What is distance ?
.................... 16
1.5
Probability and statistics
................. 22
2
Geometry of probability distributions
26
2.1
Majorization and partial order
.............. 26
2.2
Shannon entropy
...................... 32
2.3
Relative entropy
...................... 37
2.4
Continuous distributions and measures
......... 41
2.5
Statistical geometry and the Fisher-Rao metric
.... 44
2.6
Classical ensembles
.................... 49
2.7
Generalized entropies
................... 51
3
Much ado about spheres
58
3.1
Spheres
........................... 58
3.2
Parallel transport and statistical geometry
....... 62
3.3
Complex, Hermitian and
Kahler
manifolds
....... 68
3.4
Symplectic manifolds
................... 73
3.5
The
Hopf fibration
of the 3-sphere
............ 75
3.6
Fibre bundles and their connections
........... 80
3.7
The 3-sphere as a group
.................. 86
3.8
Cosets and all that
..................... 90
4
Complex
projective
spaces
94
4.1
From art to mathematics
................. 94
4.2
Complex
projective
geometry
............... 98
4.3
Complex curves, quadrics and the
Segre
embedding
. . 100
4.4
Stars, spinors and complex curves
............ 103
4.5
The Fubini-Study metric
................. 105
4.6
CP illustrated
.......................
Ill
Contents
4.7 Symplectic
geometry
and the Fubini-Study measure
. . 117
4.8
Fibre bundle aspects
.................... 118
4.9
Grassmannians and flag manifolds
............ 121
5
Outline of quantum mechanics
125
5.1
Quantum mechanics
.................... 125
5.2
Qubits and Bloch spheres
................. 127
5.3
The statistical and the Fubini-Study distances
..... 130
5.4
A real look at quantum dynamics
............ 132
5.5
Time reversals
....................... 136
5.6
Classical and quantum states: a unified approach
.... 139
6
Coherent states and group actions
144
6.1
Canonical coherent states
................. 144
6.2
Quasi-probability distributions on the plane
....... 149
6.3
Bloch coherent states
................... 156
6.4
From complex curves to SU(K) coherent states
.... 161
6.5 5*7(3)
coherent states
................... 163
7
The stellar representation
167
7.1
The stellar representation in quantum mechanics
.... 167
7.2
Orbits and coherent states
................ 169
7.3
The Husimi function
.................... 172
7.4
Wehrl entropy and the
Lieb
conjecture
......... 176
7.5
Generalized Wehrl entropies
............... 179
7.6
Random pure states
.................... 181
7.7
From the transport problem to the
Monge
distance
. . 187
8
The space of density matrices
192
8.1
Hilbert-Schmidt space and positive operators
...... 192
8.2
The set of mixed states
.................. 195
8.3
Unitary transformations
.................. 198
8.4
The space of density matrices as a convex set
...... 201
8.5
Stratification
........................ 206
8.6
An algebraic afterthought
................. 210
8.7
Summary
.......................... 212
9
Purification of mixed quantum states
214
9.1
Tensor products and state reduction
........... 214
9.2
The Schmidt decomposition
................ 217
9.3
State purification and the Hilbert-Schmidt bundle
. . . 219
9.4
A first look at the Bures metric
............. 222
9.5
Bures geometry for
N = 2................. 225
9.6
Further properties of the Bures metric
.......... 227
Contents
10 Quantum
operations
231
10.1
Measurements and POVMs
................ 231
10.2
Algebraic detour: matrix reshaping and reshuffling
. . . 238
10.3
Positive and completely positive maps
.......... 242
10.4
Environmental representations
.............. 246
10.5
Some spectral properties
................. 249
10.6
Unital and bistochastic maps
............... 250
10.7
One qubit maps
...................... 253
11
Duality: maps versus states
258
11.1
Positive and decomposable maps
............. 258
11.2
Dual cones and super-positive maps
........... 265
11.3
Jamiołkowski
isomorphism
................ 266
11.4
Quantum maps and quantum states
........... 268
12
Density matrices and entropies
273
12.1
Ordering operators
..................... 273
12.2 Von
Neumann entropy
................... 276
12.3
Quantum relative entropy
................. 282
12.4
Other entropies
....................... 285
12.5
Majorization of density matrices
............. 288
12.6
Entropy dynamics
..................... 292
13
Distinguishability measures
296
13.1
Classical distinguishability measures
........... 296
13.2
Quantum distinguishability measures
.......... 301
13.3
Fidelity and statistical distance
.............. 305
14
Monotone metrics and measures
311
14.1
Monotone metrics
..................... 311
14.2
Product measures and flag manifolds
.......... 316
14.3
Hilbert-Schmidt measure
................. 318
14.4
Bures measure
....................... 321
14.5
Induced measures
..................... 323
14.6
Random density matrices
................. 325
14.7
Random operations
.................... 328
15
Quantum entanglement
333
15.1
Introducing entanglement
................. 333
15.2
Two qubit pure states: entanglement illustrated
.... 336
15.3
Pure states of a bipartite system
............. 341
15.4
Mixed states and separability
............... 349
15.5
Geometry of the set of separable states
......... 356
15.6
Entanglement measures
.................. 361
15.7
Two-qubit mixed states
.................. 369
viii Contents
Epilogue
379
Appendix 1
Basic notions
of group theory
380
Al.l Differential forms
..................... 380
Al.
2
Riemannian curvature
................... 381
Al.3 A key fact about mappings
................ 382
Appendix
2
Basic notions of group theory
383
A2.1 Lie groups and Lie algebras
................ 383
A2.2 SU(2)
............................ 384
A2.3 SU(N)
............................ 384
A2.4 Homomorphisms between low-dimensional groups
. . . 385
Appendix
3
Geometry: do it yourself
386
Appendix
4
Hints and answers to the exercises
389
References
398
Index
416
|
| any_adam_object | 1 |
| author | Bengtsson, Ingemar Życzkowski, Karol |
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| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | repr. with corr., paperback ed. |
| format | Book |
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| id | DE-604.BV035573497 |
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| indexdate | 2024-07-09T21:40:45Z |
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| language | English |
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| spelling | Bengtsson, Ingemar Verfasser aut Geometry of quantum states an introduction to quantum entanglement Ingemar Bengtsson and Karol Życzkowski repr. with corr., paperback ed. Cambridge [u.a.] Cambridge Univ. Press 2008 XII, 419 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Quantentheorie Information theory Quantum theory Geometrie (DE-588)4020236-7 gnd rswk-swf Verschränkter Zustand (DE-588)4409616-1 gnd rswk-swf Quantenphysik (DE-588)4266670-3 gnd rswk-swf Quantenphysik (DE-588)4266670-3 s Geometrie (DE-588)4020236-7 s DE-604 Verschränkter Zustand (DE-588)4409616-1 s Życzkowski, Karol Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017629034&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bengtsson, Ingemar Życzkowski, Karol Geometry of quantum states an introduction to quantum entanglement Quantentheorie Information theory Quantum theory Geometrie (DE-588)4020236-7 gnd Verschränkter Zustand (DE-588)4409616-1 gnd Quantenphysik (DE-588)4266670-3 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4409616-1 (DE-588)4266670-3 |
| title | Geometry of quantum states an introduction to quantum entanglement |
| title_auth | Geometry of quantum states an introduction to quantum entanglement |
| title_exact_search | Geometry of quantum states an introduction to quantum entanglement |
| title_full | Geometry of quantum states an introduction to quantum entanglement Ingemar Bengtsson and Karol Życzkowski |
| title_fullStr | Geometry of quantum states an introduction to quantum entanglement Ingemar Bengtsson and Karol Życzkowski |
| title_full_unstemmed | Geometry of quantum states an introduction to quantum entanglement Ingemar Bengtsson and Karol Życzkowski |
| title_short | Geometry of quantum states |
| title_sort | geometry of quantum states an introduction to quantum entanglement |
| title_sub | an introduction to quantum entanglement |
| topic | Quantentheorie Information theory Quantum theory Geometrie (DE-588)4020236-7 gnd Verschränkter Zustand (DE-588)4409616-1 gnd Quantenphysik (DE-588)4266670-3 gnd |
| topic_facet | Quantentheorie Information theory Quantum theory Geometrie Verschränkter Zustand Quantenphysik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017629034&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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