Quantum systems, channels, information: a mathematical introduction
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
De Gruyter
2012
|
| Series: | De Gruyter Studies in Mathematical Physics
16 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [333] - 345 |
| Physical Description: | XIII, 349 S. graph. Darst. 240 mm x 170 mm |
| ISBN: | 311027325X 9783110273250 |
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MARC
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Record in the Search Index
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IMAGE 1
CONTENTS
PREFACE V
I BASIC STRUCTURES
1 VECTORS AND OPERATORS 3
1.1 HILBERT SPACE 3
1.2 OPERATORS 4
1.3 POSITIVITY 5
1.4 TRACE AND DUALITY 6
1.5 CONVEXITY 8
1.6 NOTES AND REFERENCES 9
2 STATES, OBSERVABLES, STATISTICS 10
2.1 STRUCTURE O F STATISTICAL THEORIES 10
2.1.1 CLASSICAL SYSTEMS 10
2.1.2 AXIOMS O F STATISTICAL DESCRIPTION 11
2.2 QUANTUM STATES 14
2.3 QUANTUM OBSERVABLES 16
2.3.1 QUANTUM OBSERVABLES FROM THE AXIOMS 16
2.3.2 COMPATIBILITY AND COMPLEMENTARITY 18
2.3.3 THE UNCERTAINTY RELATION 21
2.3.4 CONVEX STRUCTURE O F OBSERVABLES 22
2.4 STATISTICAL DISCRIMINATION BETWEEN QUANTUM STATES 25
2.4.1 FORMULATION O F THE PROBLEM 25
2.4.2 OPTIMAL OBSERVABLES 25
2.5 NOTES AND REFERENCES 31
3 COMPOSITE SYSTEMS AND ENTANGLEMENT 34
3.1 COMPOSITE SYSTEMS 34
3.1.1 TENSOR PRODUCTS 34
3.1.2 NAIMARK'S DILATION 36
3.1.3 SCHMIDT DECOMPOSITION AND PURIFICATION 38
HTTP://D-NB.INFO/1027078265
IMAGE 2
X
CONTENTS
3.2 QUANTUM ENTANGLEMENT VS "LOCAL REALISM" 41
3.2.1 PARADOX O F EINSTEIN-PODOLSKI-ROSEN AND BELL'S INEQUALITIES . . 41
3.2.2 MERMIN-PERES GAME 45
3.3 QUANTUM SYSTEMS AS INFORMATION CARRIERS 47
3.3.1 TRANSMISSION O F CLASSICAL INFORMATION 47
3.3.2 ENTANGLEMENT AND LOCAL OPERATIONS 48
3.3.3 SUPERDENSE CODING 4 9
3.3.4 QUANTUM TELEPORTATION 50
3.4 NOTES AND REFERENCES 52
II THE PRIMARY CODING THEOREMS
4 CLASSICAL ENTROPY AND INFORMATION 57
4.1 ENTROPY O F A RANDOM VARIABLE AND DATA COMPRESSION 57
4.2 CONDITIONAL ENTROPY AND THE SHANNON INFORMATION 59
4.3 THE SHANNON CAPACITY O F THE CLASSICAL NOISY CHANNEL 62
4.4 THE CHANNEL CODING THEOREM 64
4.5 WIRETAP CHANNEL 69
4.6 GAUSSIAN CHANNEL 71
4.7 NOTES AND REFERENCES 72
5 THE CLASSICAL-QUANTUM CHANNEL 74
5.1 CODES AND ACHIEVABLE RATES 74
5.2 FORMULATION O F THE CODING THEOREM 75
5.3 THE UPPER BOUND 78
5.4 PROOF O F THE WEAK CONVERSE 83
5.5 TYPICAL PROJECTORS 87
5.6 PROOF O F THE DIRECT CODING THEOREM 92
5.7 THE RELIABILITY FUNCTION FOR PURE-STATE CHANNEL 95
5.8 NOTES AND REFERENCES 98
III CHANNELS AND ENTROPIES
6 QUANTUM EVOLUTIONS AND CHANNELS 103
6.1 QUANTUM EVOLUTIONS 103
6.2 COMPLETELY POSITIVE MAPS 106
6.3 DEFINITION O F THE CHANNEL 112
IMAGE 3
CONTENTS XI
6.4 ENTANGLEMENT-BREAKING AND PPT CHANNELS 114
6.5 QUANTUM MEASUREMENT PROCESSES 117
6.6 COMPLEMENTARY CHANNELS 119
6.7 COVARIANT CHANNELS 124
6.8 QUBIT CHANNELS 127
6.9 NOTES AND REFERENCES 129
7 QUANTUM ENTROPY AND INFORMATION QUANTITIES 132
7.1 QUANTUM RELATIVE ENTROPY 132
7.2 MONOTONICITY O F THE RELATIVE ENTROPY 133
7.3 STRONG SUBADDITIVITY O F THE QUANTUM ENTROPY 138
7.4 CONTINUITY PROPERTIES 140
7.5 INFORMATION CORRELATION, ENTANGLEMENT O F FORMATION AND CONDITIONAL
ENTROPY 142
7.6 ENTROPY EXCHANGE 147
7.7 QUANTUM MUTUAL INFORMATION 149
7.8 NOTES AND REFERENCES 151
IV BASIC CHANNEL CAPACITIES
8 THE CLASSICAL CAPACITY O F QUANTUM CHANNEL 155
8.1 THE CODING THEOREM 155
8.2 THE /-CAPACITY 157
8.3 THE ADDITIVITY PROBLEM 160
8.3.1 THE EFFECT O F ENTANGLEMENT IN ENCODING AND DECODING 160
8.3.2 A HIERARCHY O F ADDITIVITY PROPERTIES 164
8.3.3 SOME ENTROPY INEQUALITIES 166
8.3.4 ADDITIVITY FOR COMPLEMENTARY CHANNELS 169
8.3.5 NONADDITIVITY OF QUANTUM ENTROPY QUANTITIES 171
8.4 NOTES AND REFERENCES 178
9 ENTANGLEMENT-ASSISTED CLASSICAL COMMUNICATION 180
9.1 THE GAIN O F ENTANGLEMENT ASSISTANCE 180
9.2 THE CLASSICAL CAPACITIES OF QUANTUM OBSERVABLES 184
9.3 PROOF O F THE CONVERSE CODING THEOREM 188
9.4 PROOF O F THE DIRECT CODING THEOREM 190
9.5 NOTES AND REFERENCES 194
IMAGE 4
XII CONTENTS
10 TRANSMISSION O F QUANTUM INFORMATION 195
10.1 QUANTUM ERROR-CORRECTING CODES 195
10.1.1 ERROR CORRECTION BY REPETITION 195
10.1.2 GENERAL FORMULATION 197
10.1.3 NECESSARY AND SUFFICIENT CONDITIONS FOR ERROR CORRECTION 198
10.1.4 COHERENT INFORMATION AND PERFECT ERROR CORRECTION 200
10.2 FIDELITIES FOR QUANTUM INFORMATION 203
10.2.1 FIDELITIES FOR PURE STATES 203
10.2.2 RELATIONS BETWEEN THE FIDELITY MEASURES 205
10.2.3 FIDELITY AND THE BURES DISTANCE 208
10.3 THE QUANTUM CAPACITY 210
10.3.1 ACHIEVABLE RATES 210
10.3.2 THE QUANTUM CAPACITY AND THE COHERENT INFORMATION 215
10.3.3 DEGRADABLE CHANNELS 217
10.4 THE PRIVATE CLASSICAL CAPACITY AND THE QUANTUM CAPACITY 220
10.4.1 THE QUANTUM WIRETAP CHANNEL 220
10.4.2 PROOF O F THE PRIVATE CAPACITY THEOREM 223
10.4.3 LARGE DEVIATIONS FOR RANDOM OPERATORS 229
10.4.4 THE DIRECT CODING THEOREM FOR THE QUANTUM CAPACITY 232
10.5 NOTES AND REFERENCES 237
V INFINITE S Y S T E M S
11 CHANNELS WITH CONSTRAINED INPUTS 243
11.1 CONVERGENCE OF DENSITY OPERATORS 243
11.2 QUANTUM ENTROPY AND RELATIVE ENTROPY 247
11.3 CONSTRAINED C-Q CHANNEL 249
11.4 CLASSICAL-QUANTUM CHANNEL WITH CONTINUOUS ALPHABET 252
11.5 CONSTRAINED QUANTUM CHANNEL 254
11.6 ENTANGLEMENT-ASSISTED CAPACITY O F CONSTRAINED CHANNELS 257
11.7 ENTANGLEMENT-BREAKING CHANNELS IN INFINITE DIMENSIONS 259
11.8 NOTES AND REFERENCES 264
12 GAUSSIAN SYSTEMS 266
12.1 PRELIMINARY MATERIAL 266
12.1.1 SPECTRAL DECOMPOSITION AND STONE'S THEOREM 266
12.1.2 OPERATORS ASSOCIATED WITH THE HEISENBERG COMMUTATION RELATION 269
IMAGE 5
CONTENTS XIII
12.1.3 CLASSICAL SIGNAL PLUS QUANTUM NOISE 272
12.1.4 THE CLASSICAL-QUANTUM GAUSSIAN CHANNEL 275
12.2 CANONICAL COMMUTATION RELATIONS 276
12.2.1 WEYL-SEGAL CCR 276
12.2.2 THE SYMPLECTIC SPACE 279
12.2.3 DYNAMICS, QUADRATIC OPERATORS AND GAUGE TRANSFORMATIONS . . . .
281
12.3 GAUSSIAN STATES 284
12.3.1 CHARACTERISTIC FUNCTION 284
12.3.2 DEFINITION AND PROPERTIES O F GAUSSIAN STATES 285
12.3.3 THE DENSITY OPERATOR O F GAUSSIAN STATE 289
12.3.4 ENTROPY O F A GAUSSIAN STATE 290
12.3.5 SEPARABILITY AND PURIFICATION 293
12.4 GAUSSIAN CHANNELS 296
12.4.1 OPEN BOSONIC SYSTEMS 296
12.4.2 GAUSSIAN CHANNELS: BASIC PROPERTIES 300
12.4.3 GAUSSIAN OBSERVABLES 301
12.4.4 GAUSSIAN ENTANGLEMENT-BREAKING CHANNELS 303
12.5 THE CAPACITIES O F GAUSSIAN CHANNELS 307
12.5.1 MAXIMIZATION O F THE MUTUAL INFORMATION 307
12.5.2 GAUGE-COVARIANT CHANNELS 308
12.5.3 MAXIMIZATION O F THE COHERENT INFORMATION 310
12.5.4 THE CLASSICAL CAPACITY: CONJECTURES 311
12.6 THE CASE OF ONE MODE 314
12.6.1 CLASSIFICATION O F GAUSSIAN CHANNELS 314
12.6.2 ENTANGLEMENT-BREAKING CHANNELS 320
12.6.3 ATTENUATION/AMPLIFICATION/CLASSICAL NOISE CHANNEL 321
12.6.4 ESTIMATING THE QUANTUM CAPACITY 325
12.7 NOTES AND REFERENCES 329
BIBLIOGRAPHY 333
INDEX 346 |
| any_adam_object | 1 |
| author | Cholevo, Aleksandr S. 1943- |
| author_GND | (DE-588)12282735X |
| author_facet | Cholevo, Aleksandr S. 1943- |
| author_role | aut |
| author_sort | Cholevo, Aleksandr S. 1943- |
| author_variant | a s c as asc |
| building | Verbundindex |
| bvnumber | BV040613074 |
| classification_rvk | ST 152 UG 3000 UK 8000 |
| ctrlnum | (OCoLC)820309196 (DE-599)DNB1027078265 |
| dewey-full | 530.15 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.15 |
| dewey-search | 530.15 |
| dewey-sort | 3530.15 |
| dewey-tens | 530 - Physics |
| discipline | Physik Informatik |
| format | Book |
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| indexdate | 2024-08-21T00:22:49Z |
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| isbn | 311027325X 9783110273250 |
| language | English |
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| physical | XIII, 349 S. graph. Darst. 240 mm x 170 mm |
| publishDate | 2012 |
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| spelling | Cholevo, Aleksandr S. 1943- Verfasser (DE-588)12282735X aut Quantum systems, channels, information a mathematical introduction Alexander S. Holevo Berlin [u.a.] De Gruyter 2012 XIII, 349 S. graph. Darst. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter Studies in Mathematical Physics 16 Literaturverz. S. [333] - 345 Quanteninformatik (DE-588)4705961-8 gnd rswk-swf Quanteninformatik (DE-588)4705961-8 s DE-604 Erscheint auch als Online-Ausgabe 978-3-11-027340-3 De Gruyter Studies in Mathematical Physics 16 (DE-604)BV040141722 16 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4165194&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025440596&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cholevo, Aleksandr S. 1943- Quantum systems, channels, information a mathematical introduction De Gruyter Studies in Mathematical Physics Quanteninformatik (DE-588)4705961-8 gnd |
| subject_GND | (DE-588)4705961-8 |
| title | Quantum systems, channels, information a mathematical introduction |
| title_auth | Quantum systems, channels, information a mathematical introduction |
| title_exact_search | Quantum systems, channels, information a mathematical introduction |
| title_full | Quantum systems, channels, information a mathematical introduction Alexander S. Holevo |
| title_fullStr | Quantum systems, channels, information a mathematical introduction Alexander S. Holevo |
| title_full_unstemmed | Quantum systems, channels, information a mathematical introduction Alexander S. Holevo |
| title_short | Quantum systems, channels, information |
| title_sort | quantum systems channels information a mathematical introduction |
| title_sub | a mathematical introduction |
| topic | Quanteninformatik (DE-588)4705961-8 gnd |
| topic_facet | Quanteninformatik |
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