Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems:
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2011
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| 245 | 1 | 0 | |a Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems |c Masanori Ohya ; Igor Volovich |
| 264 | 1 | |a Dordrecht [u.a.] |b Springer |c 2011 | |
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Titel: Mathematical foundations of quantum information and computation and its applications to nano- and bi
Autor: Ohya, Masanori
Jahr: 2011
Contents
1 Introduction. 1
1.1 Entropy and Information Communication. 2
1.2 Classical Computer Versus Quantum Computer. 4
1.2.1 Logical Irreversibility and Energy Loss. 4
1.2.2 Thermodynamic Interpretation of Irreversible Computation 5
1.2.3 Information Theoretic Interpretation. 5
1.2.4 Resolution of Demerits (1), (2) by Quantum Computer . . 5
1.3 What Are Quantum Gates and Computation? . 6
1.3.1 Gates. 6
1.3.2 Quantum Algorithm and Computation. 8
1.4 Locality and Entanglement. 9
1.5 What Are Quantum Cryptography and Teleportation?. 11
1.5.1 Cryptography. 11
1.5.2 Teleportation. 12
1.6 Nanosystems and Biosystems. 12
2 Algorithms and Computation. 15
2.1 General Algorithms. 15
2.2 Turing Machine. 17
2.2.1 Gödel Encoding. 19
2.2.2 The Halting Problem. 21
2.2.3 Universal Turing Machine. 21
2.2.4 Partially Recursive Functions. 21
2.3 Boolean Functions, Gates and Circuits. 23
2.3.1 Boolean Functions and Gates. 23
2.3.2 Circuits. 25
2.4 Computational Complexity, P-problem and NP-problem . 26
2.5 Notes. 28
3 Basics of Classical Probability. 29
3.1 Classical Probability. 29
3.1.1 Radon-Nikodym Theorem. 32
ix
Contents
3.2 Conditional Expectation and Martingale. 33
3.3 Algebraic Formulation of Classical Probability . 34
3.4 Notes. 35
Basics of Infinite-Dimensional Analysis. 37
4.1 Hubert Space. 37
4.2 Linear Operators in Hubert Space. 39
4.2.1 Spaces F(W),C(W),S(tt), and T(H). 41
4.3 Direct Sum and Tensor Product of Hubert Spaces. 42
4.4 Fock Space. 44
4.4.1 Bosonic Fock Space. 44
4.4.2 Annihilation and Creation Operators . 45
4.4.3 Fermionic Fock Space. 46
4.4.4 Weyl Representation. 47
4.5 Hida White Noise Analysis. 48
4.5.1 White Noise. 48
4.5.2 Hida Distributions. 49
4.5.3 Lie Algebra of the Hida Rotation Group. 51
4.6 Elements of Operator Algebras. 52
4.7 KMS States and Tomita-Takesaki Theory. 56
4.8 Notes. 59
Basics of Quantum Mechanics. 61
5.1 Quantum Dynamics. 61
5.1.1 Schrödinger Picture. 62
5.1.2 Heisenberg Picture. 63
5.1.3 FreeParticle. 63
5.1.4 Scattering. 65
5.1.5 Description of General State: Density Operator. 65
5.2 States and Observables. 67
5.2.1 Quantum Parallelism (Duality) . 67
5.2.2 States. 70
5.2.3 Observables . 70
5.2.4 Expectation Values. 71
5.3 Quantum Oscillator, Creation and Annihilation Operators. 72
5.4 Symmetries. 73
5.4.1 Representations of the Rotation Group: Spin. 74
5.5 Quantum Probability. 75
5.5.1 Conditional Probability and Joint Probability in QP . . 77
5.5.2 Probability Positive Operator-Valued Measure (POVM) . 80
5.6 General Frameworks of Quantum Probability. 82
5.7 Uncertainty Relation. 82
5.8 Principle of Quantum Measurements. 83
5.8.1 Measurement Procedure. 84
5.8.2 Reductionof State. 85
5.9 Composite Systems and Operations . 88
Contents
5.9.1 Operations and Completely Positive Mappings. 89
5.9.2 Measurements as Unitary Transformations on Extended
Space. 90
5.10 POVM and Instruments. 91
5.10.1 POVM. 91
5.10.2 Instrument. 92
5.10.3 Covariant POVM and Instrument. 94
5.11 Seven Principles of Quantum Mechanics. 95
5.12 Notes. 99
Fundamentals of Classical Information Theory. 101
6.1 Communication Processes and Channel. 101
6.2 Entropy. 103
6.3 Relative Entropy . 108
6.4 Mutual Entropy. 111
6.4.1 Discrete Case. 111
6.4.2 Continuous Case. 112
6.5 Entropy of Information Source. 114
6.6 Capacity. 116
6.7 McMillan's Theorem. 117
6.8 Shannon's Coding Theorem . 118
6.8.1 Channel, Transmission Rate and Capacity Rate. 119
6.8.2 Coding Theorem. 121
6.8.3 Interpretation of Coding Theorem. 122
6.9 Notes. 123
Fundamentals of Quantum Information. 125
7.1 Communication Processes. 126
7.2 Quantum Entropy for Density Operators. 127
7.3 Relative Entropy for Density Operators . 131
7.4 More About Channels. 136
7.4.1 von Neumann Entropy for the Qubit State. 142
7.5 Quantum Mutual Entropy. 143
7.5.1 Generalized (or Quasi) Quantum Mutual Entropy. 147
7.5.2 Ergodic Type Theorem. 148
7.6 Entropy Exchange and Coherent Information . 149
7.7 Comparison of Various Quantum Mutual Type Entropies. 150
7.8 Lifting and Beam Splitting. 152
7.9 Entropies for General Quantum States. 158
7.10 Sufficiency and Relative Entropy. 163
7.11 Notes. 165
Locality and Entanglement. 167
8.1 EPR Model and Böhm Model. 168
8.1.1 Various Localities. 168
8.1.2 Probability and Local Causality. 170
xii Contents
8.1.3 EPR Model vs. Böhm and Bell Model. 172
8.1.4 Bell's Locality. 173
8.1.5 Discussion of Bell's Locality . 174
8.1.6 Example of Local Realist Representation for Spins . 174
8.1.7 Local Realist Representation for EPR Correlations . 175
8.1.8 On Bell's Theorem. 178
8.2 Bell's Theorem. 181
8.2.1 Bell's Theorem and Stochastic Processes. 181
8.2.2 CHSH Inequality. 182
8.3 Various Local Realisms. 184
8.3.1 Bell's Local Realism. 184
8.3.2 Space and Time in Axioms of Quantum Mechanics . . . . 185
8.3.3 Einstein's Local Realism. 185
8.3.4 Local Realistic Representation for Quantum Spin
Correlations . 186
8.3.5 Correlation Functions and Local Realism. 186
8.4 Entangled States in Space-Time. Disentanglement . 188
8.5 Local Observations. 190
8.5.1 Modified Bell's Equation . 190
8.6 Separability and Entanglement. 193
8.6.1 Entangling Operator. 194
8.6.2 True Quantum Entanglement, d- and c-Entanglements . . 198
8.6.3 Criteria of Entangled States. 203
8.6.4 Degrees of Entanglement . 209
8.6.5 Models of Entanglement in Circulant States. 215
8.6.6 Computation of DEN . 216
8.7 Entangled Markov Chains and Their Properties. 218
8.7.1 Entangled Quantum Markov Chains. 218
8.7.2 Entangled Markov Chains Generated by Unitary
Implementable Matrices. 222
8.8 Notes. 225
9 Quantum Capacity and Coding. 227
9.1 Channel Capacity. 227
9.1.1 Capacity of Quantum Channel. 227
9.1.2 Capacity of Classical-Quantum-Classical Channel . 229
9.1.3 Boundof Mutual Entropy and Capacity. 229
9.2 Computation of Capacity. 233
9.2.1 Divergence Center. 235
9.2.2 Comparison of Capacities. 238
9.3 Quantum McMillan Type Theorem. 241
9.3.1 Entropy Operators in Quantum Systems. 241
9.3.2 McMillan's Type Convergence Theorems . 243
9.4 Coding Type Theorems. 246
9.5 Notes. 250
Contents xiii
10 Information Dynamics and Adaptive Dynamics.251
10.1 Complex Systems.251
10.2 Information Dynamics.252
10.3 State Change and Complexities.254
10.4 Adaptive Dynamics.260
10.4.1 Entropy, Information in Classical and Quantum World . . 262
10.4.2 Schematic Expression of Understanding.262
10.5 Adaptive Dynamics?Conceptual Meaning.263
10.5.1 Description of Chaos.263
10.5.2 Chameleon Dynamics.264
10.5.3 Quantum SAT Algorithm.264
10.5.4 Summary of Adaptive Dynamics .266
10.6 AUseofID: Chaos Degree.267
10.7 Algorithm Computing Entropie Chaos Degree (AUseof AD) . . 269
10.7.1 LogisticMap.270
10.7.2 ECD with Memory.279
10.8 Adaptive Dynamics Describing Chaos.282
10.8.1 Chaos Degree with Adaptivity.284
10.9 Time Irreversibility Problem and Functional Mechanics.288
10.9.1 States and Observables in Functional Classical Mechanics 292
10.9.2 Free Motion.295
10.9.3 Newton's Equation for the Average Coordinate.295
10.9.4 Comparison with Quantum Mechanics.296
10.9.5 Liouville Equation and the Newton Equation.297
10.9.6 Corrections to Newton's Equations.299
10.9.7 TimeReversal.301
10.9.8 Dynamics of a Particle in a Box.304
10.10 New Interpretation of Bell's Inequality?Chameleon Dynamics . . 306
10.10.1 Dynamical Systems: Passive and Adaptive. 306
10.10.2 The EPRB-Chameleon Dynamical System. 308
10.10.3 Probabilistic Error Model. 310
10.10.4 Upper Bounds for CHSH Inequality. 311
10.11 Notes. 312
11 Mathematical Models of Quantum Computer.313
11.1 Quantum Turing Machine. 313
11.1.1 Universal Quantum Turing Machine. 316
11.2 Quantum Gates. 319
11.3 Quantum Circuits. 321
11.4 Universal Quantum Gates. 323
11.5 Problem on the Halting Scheme . 324
11.5.1 Destruction of Computation Result.324
11.5.2 QND (Quantum Non-demolition Monitoring)-Type
Quantum Turing Machine.325
11.5.3 Problem for Halting.325
xiv Contents
11.6 Generalized Quantum Turing Machine.327
11.7 Notes .329
12 Quantum Algorithms I.331
12.1 Discrete Quantum Fourier Transform and Principle of Quantum
Algorithm.331
12.1.1 HadamardGate. 331
12.1.2 Discrete Quantum Fourier Transformation . 331
12.1.3 Principle of Quantum Algorithm. 334
12.2 Deutsch-Jozsa Algorithm. 334
12.3 Grover's Search Algorithm. 336
12.3.1 Complexity of Grover's Algorithm .343
12.3.2 Accardi-Sabadini's Unitary Operator.350
12.4 Notes.354
13 Quantum Algorithms II.355
13.1 Elements of Number Theory.355
13.1.1 Euclid's Algorithm. 355
13.1.2 The Euler Function. 359
13.1.3 Modular Exponentiation. 360
13.2 Shor's Quantum Factoring. 361
13.2.1 Finding the Order. 362
13.2.2 Description of the Algorithm . 363
13.2.3 Computational Complexity of Shor's Algorithm. 365
13.3 Factoring Integers. 366
13.3.1 Factoring Algorithm. 366
13.4 Notes. 368
14 Quantum Algorithm III.369
14.1 Quantum Algorithm of SAT Problem.370
14.1.1 Quantum Computing. 371
14.1.2 Quantum Algorithm of SAT Problem. 374
14.1.3 Example. 378
14.2 Quantum Chaos Algorithm. 380
14.2.1 Amplification Process in SAT Algorithm.381
14.2.2 Computational Complexity of SAT Algorithm.384
14.3 Channel Expression of Quantum Algorithm.385
14.3.1 Channel Expression of Unitary Algorithm.385
14.3.2 Channel Expression of Arbitrary Algorithm.386
14.4 SAT Algorithm in GQTM.387
14.5 SAT Algorithm with Stochastic Limits.392
14.6 Notes.395
15 Quantum Error Correction.397
15.1 Three Qubit Code and Fidelity.397
15.1.1 Three Qubit Code.397
Contents xv
15.1.2 Fidelity and Error Correction . 399
15.2 The Shor Code. 402
15.3 Calderbank-Shor-Steane Codes. 403
15.4 General Theory of Quantum Error-Correcting Codes. 404
15.5 Depolarizing Channel. 405
15.6 Fault-Tolerant Quantum Computation and the Threshold Theorem 406
15.7 Notes. 412
16 Quantum Field Theory, Locality and Entanglement. 413
16.1 Quantum Electrodynamics (QED). 413
16.1.1 Maxwell Equations. 414
16.1.2 Quantization of Electromagnetic Field . 415
16.1.3 Casimir Effect. 416
16.1.4 Correlation Functions and Photo-Detection. 418
16.1.5 Interference: Two-Slits Experiment. 419
16.1.6 Lorentz-Invariant Form of Maxwell Equations. 420
16.1.7 Dirac Equation. 421
16.1.8 Pauli Equation. 422
16.1.9 Equations of Quantum Electrodynamics. 423
16.2 Quantum Fields and Locality. 423
16.2.1 Wightman Axioms. 423
16.2.2 Algebraic Quantum Theory and Local Channels. 424
16.3 Quantum Field Theory in Quantum Probability Scheme. 425
16.4 Expansion of Wave Packet. 428
16.4.1 Relativistic Particles. 428
16.5 Space Dependence of the Dirac Correlation Function. 429
16.6 WavePackets. 432
16.6.1 Calculations and Results. 434
16.7 Noncommutative Spectral Theory and Local Realism. 437
16.8 Contextual Approach. 439
16.8.1 Contextual Classical and Quantum Probability. 439
16.9 Notes. 440
17 Quantum Cryptography. 441
17.1 Private Key Cryptosystems. 442
17.1.1 Julius Caesar's Cryptosystem. 442
17.1.2 Symmetrie Cryptosystems?DES and GOST. 443
17.2 Public Key Cryptography and RSA Cryptosystem. 443
17.2.1 The RSA Protocol. 444
17.2.2 Mathematical Basis of the RSA Protocol. 445
17.3 Entropie Uncertainty Relations. 446
17.4 No-cloning Theorem. 447
17.5 The BB84 Quantum Cryptographic Protocol. 449
17.5.1 The BB84 Protocol. 449
17.5.2 TheSecurityofBB84. 450
xvi Contents
17.5.3 Ultimate Security Proofs.451
17.6 The EPRBE Quantum Cryptographic Protocol.452
17.6.1 Quantum Nonlocality and Cryptography .452
17.6.2 Bell's Inequality and Localized Detectors.453
17.6.3 The EPRBE Quantum Key Distribution.454
17.6.4 Gaussian Wave Functions.455
17.7 Notes.457
18 Quantum Teleportation.459
18.1 Channel Expression of Quantum Teleportation.460
18.2 BBCJPW Model of Teleportation.462
18.3 Weak Teleportation and Uniqueness of Key.463
18.4 Perfect Teleportation in Böse Fock Space.466
18.4.1 Basic Notations and Facts. 467
18.4.2 A Perfect Teleportation . 473
18.5 Non-perfect Teleportation in Böse Fock Space. 475
18.6 Fidelity in Teleportation . 481
18.7 Spatially Separated Teleportation. 486
18.7.1 Teleportation of States InsideR* .486
18.7.2 Alice and Bob Are Spatially Separated.489
18.8 Model of Continuous Teleportation.491
18.8.1 Scheme of Continuous Variable Teleportation.491
18.8.2 Entangled State Employed by Continuous Teleportation . 494
18.9 Quantum Teleportation with Non-maximally Entangled States . . 495
18.9.1 Basic Setting.496
18.9.2 New Scheme of Teleportation.498
18.9.3 Examples.501
18.9.4 Perfect Teleportation for Non-maximally Entangled State 502
18.10 Notes.504
19 Physical Nanosystems.505
19.1 Quantum Dots.505
19.1.1 Quantum Wells, Quantum Wires, Quantum Dots.505
19.1.2 Fock-Darwin Hamiltonian and Interacting Electrons in
Quantum Dot.506
19.1.3 Properties of Quantum Dots.508
19.1.4 Quantum Computation with Quantum Dots.508
19.2 Quantum Communication Experiments .509
19.2.1 Quantum Cryptography and Teleportation.509
19.3 Experimental Realizations of Quantum Computations.510
19.3.1 Ion Traps. 510
19.3.2 Nuclear Magnetic Resonance. 512
19.3.3 Atomic Quantum Computer. 514
19.4 Parametric Down Conversion. 516
19.4.1 Notes. 519
Contents xvii
20 Miscellaneous in Quantum Theory and Information.521
20.1 Lifting and Conditional Expectation.521
20.1.1 Lifting. 521
20.1.2 Conditional Probability . 522
20.1.3 Various Liftings. 525
20.2 Various Topics on Entropies . 528
20.2.1 Boltzmann's Entropy. 528
20.2.2 Entropy Increase. 528
20.2.3 Gibbs Entropy. 529
20.2.4 Boltzmann Entropy in Quantum Mechanics. 530
20.2.5 von Neumann Microscopic and Macroscopic Entropies . . 531
20.2.6 Entropy Change for Open Systems . 531
20.2.7 Reversible and Irreversible Processes. 534
20.2.8 Entropy Production . 535
20.3 Quantum Dynamical Entropy. 539
20.3.1 Formulation by CNT.539
20.3.2 Kolmogorov-Sinai Type Complexities .539
20.3.3 Model Computation.541
20.3.4 Various Quantum Dynamical Entropies.543
20.3.5 Some Models.553
20.4 Fractal Dimension of State.558
20.4.1 Applications.560
20.5 Entanglement in Janes-Cummings Model.562
20.6 Decoherence in Quantum Systems: Stochastic Limit.565
20.6.1 Reducing of Decoherence in Quantum Computers . 566
20.6.2 Stochastic Limit.567
20.7 Properties of the Quantum Baker's Map.571
20.7.1 Quantum vs. Classical Dynamics. 573
20.7.2 Coherent States for the Quantum Baker's Map. 575
20.7.3 Expectation Values and Chaos Degree. 578
20.7.4 Numerica! Simulation of the Chaos Degree and
Classical-Quantum Correspondence. 591
20.7.5 Quantum-Classical Correspondence for Quantum
Baker's Map.594
20.8 Combined Quantum Baker's Map .598
20.8.1 A Combined Classical Baker's Map and Its Entropie
Chaos Degree .598
20.8.2 A Combined Quantum Baker's Map and Its Entropie
Chaos Degree . 599
20.8.3 Dependence of the Entropie Chaos Degree on the
Combination Parameter a. 601
20.8.4 Dependence of the Entropie Chaos Degree on N. 604
20.9 Notes. 605
21 Applications to Life Science.607
21.1 Introduction.608
Contents
21.1.1 Quantum Theory and Molecular Biology.608
21.1.2 On Quantum Mind and Brain.609
21.1.3 Characteristics of Biological Systems.610
21.2 Genome and Proteins.612
21.2.1 Cell. 612
21.2.2 Biomolecules. 613
21.2.3 Nucleic Acids (DNA and RNA). 614
21.2.4 GeneticCode. 615
21.2.5 Human Genome. 617
21.3 Sequence Alignment by Quantum Algorithm . 618
21.3.1 Outline of Alignment . 618
21.3.2 Quantum Algorithm of Multiple Alignment. 619
21.3.3 Computational Complexity of Quantum Algorithm . 625
21.4 A Brain Model Describing the Recognition Process. 625
21.4.1 A Quantum-Like Model of Brain. 626
21.4.2 Value of Information in Brain. 627
21.4.3 Postulates Concerning the Recognition of Signals . 628
21.4.4 The Space of Signals. 629
21.4.5 The Memory Space . 631
21.4.6 The Exchange Operator. 632
21.4.7 Processing of the Signals. 634
21.4.8 Recognition of Signals. 634
21.4.9 MeasurementsoftheEEG-Type. 636
21.5 Evolution Tree and Study of the HIV and the Influenza A Viruses:
As Applications of Information Dynamics. 636
21.5.1 Genetic Difference and Evolution Tree. 637
21.5.2 A Method Constructing Phylogenetic Trees. 638
21.5.3 Phylogenetic Tree for Hemoglobin a. 643
21.5.4 Evolution and Its Tree of HIV byEER . 645
21.5.5 Use of the Entropie Chaos Degree to Study Influenza A
Viruses and HI V-l. 647
21.5.6 Entropie Chaos Degree. 651
21.5.7 Evolution Analysis of Influenza A Viruses . 652
21.5.8 Evolution of the HIV-1 Viruses byECD. 658
21.6 Code Structureof HIV-1 . 660
21.6.1 Coding Theory.660
21.6.2 Application of Artificial Codes to Genome.662
21.6.3 Code Structureof HI V-l.665
21.7 p-adic Modeling of the Genome and the Genetic Code.668
21.7.1 GeneticCode.670
21.7.2 p-adic Genome.672
21.7.3 p-adic Genetic Code.678
21.7.4 Remarks.685
21.8 Mathematical Model of Drag Delivery System.687
21.8.1 Mathematical Model.688
Contents xix
21.8.2 Results and Discussion. 693
21.9 An Application of Quantum Information to a Biosystem:
Biomolecular Modeling. 697
21.9.1 Folding Problem. 697
21.9.2 Molecular Dynamics. 698
21.9.3 Molecular Dynamics for Harmonie Oscillator. 701
21.9.4 Examples of Molecular Dynamics Simulations. 703
21.9.5 Quantum Algorithm of Protein Folding. 704
21.10 Quantum Photosynthesis. 704
21.10.1 Photosystems. 705
21.10.2 Biophysicsof Photosynthesis. 706
21.10.3 Quantum Mechanics in Photosynthesis. 707
21.10.4 Quantum Network Model. 709
21.10.5 Chaotic Amplifier and Decreasingof Entropy. 710
21.10.6 Channel Representation of Photosynthesis . 711
21.11 Quantum-Like Models for Cognitive Psychology. 712
21.11.1 Quantum-Like Model for Decision-Making in
Two-Player Game. 712
21.11.2 Decision-Making Process in Player's Mind. 713
21.11.3 Definition of Velocities of £ and k. 714
21.11.4 Decision-Making in PD-Type Game and Irrational Choice 717
21.11.5 Non-Kolmogorovian Structure. 718
References. 719
Index. 751 |
| any_adam_object | 1 |
| author | Ohya, Masanori 1947- Volovič, Igorʹ V. 1946- |
| author_GND | (DE-588)133939693 (DE-588)123091411 |
| author_facet | Ohya, Masanori 1947- Volovič, Igorʹ V. 1946- |
| author_role | aut aut |
| author_sort | Ohya, Masanori 1947- |
| author_variant | m o mo i v v iv ivv |
| building | Verbundindex |
| bvnumber | BV037289411 |
| classification_rvk | UK 1200 |
| ctrlnum | (OCoLC)712591230 (DE-599)DNB1005845069 |
| discipline | Physik |
| format | Book |
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| id | DE-604.BV037289411 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:02:00Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021202035 |
| oclc_num | 712591230 |
| open_access_boolean | |
| owner | DE-11 |
| owner_facet | DE-11 |
| physical | XIX, 759 S. graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series2 | Theoretical and mathematical physics |
| spelling | Ohya, Masanori 1947- Verfasser (DE-588)133939693 aut Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems Masanori Ohya ; Igor Volovich Dordrecht [u.a.] Springer 2011 XIX, 759 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Theoretical and mathematical physics Quantencomputer (DE-588)4533372-5 gnd rswk-swf Quanteninformatik (DE-588)4705961-8 gnd rswk-swf Quantencomputer (DE-588)4533372-5 s Quanteninformatik (DE-588)4705961-8 s DE-604 Volovič, Igorʹ V. 1946- Verfasser (DE-588)123091411 aut X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3524317&prov=M&dok_var=1&dok_ext=htm Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021202035&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ohya, Masanori 1947- Volovič, Igorʹ V. 1946- Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems Quantencomputer (DE-588)4533372-5 gnd Quanteninformatik (DE-588)4705961-8 gnd |
| subject_GND | (DE-588)4533372-5 (DE-588)4705961-8 |
| title | Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems |
| title_auth | Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems |
| title_exact_search | Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems |
| title_full | Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems Masanori Ohya ; Igor Volovich |
| title_fullStr | Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems Masanori Ohya ; Igor Volovich |
| title_full_unstemmed | Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems Masanori Ohya ; Igor Volovich |
| title_short | Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems |
| title_sort | mathematical foundations of quantum information and computation and its applications to nano and bio systems |
| topic | Quantencomputer (DE-588)4533372-5 gnd Quanteninformatik (DE-588)4705961-8 gnd |
| topic_facet | Quantencomputer Quanteninformatik |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3524317&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021202035&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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