Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems (eBook)

M. Ohya (Tokyo University of Science) and I. Volovich (SMI RAS, Russia) are acknowledged experts on quantum information theory.

Preface 6
Acknowledgements 8
Contents 9
Introduction 20
Entropy and Information Communication 21
Classical Computer Versus Quantum Computer 23
Logical Irreversibility and Energy Loss 23
Thermodynamic Interpretation of Irreversible Computation 24
Information Theoretic Interpretation 24
Resolution of Demerits (1), (2) by Quantum Computer 24
What Are Quantum Gates and Computation? 25
Gates 25
Quantum Algorithm and Computation 27
Locality and Entanglement 28
What Are Quantum Cryptography and Teleportation? 30
Cryptography 30
Teleportation 31
Nanosystems and Biosystems 31
Algorithms and Computation 33
General Algorithms 33
Turing Machine 35
Gödel Encoding 37
The Halting Problem 39
Universal Turing Machine 39
Partially Recursive Functions 39
Boolean Functions, Gates and Circuits 41
Boolean Functions and Gates 41
Circuits 43
Computational Complexity, P-problem and NP-problem 44
Notes 46
Basics of Classical Probability 47
Classical Probability 47
Radon-Nikodym Theorem 50
Conditional Expectation and Martingale 51
Algebraic Formulation of Classical Probability 52
Notes 53
Basics of Infinite-Dimensional Analysis 54
Hilbert Space 54
Linear Operators in Hilbert Space 56
Spaces F(H),C(H),S(H), and T(H) 58
Direct Sum and Tensor Product of Hilbert Spaces 59
Fock Space 61
Bosonic Fock Space 61
Annihilation and Creation Operators 62
Fermionic Fock Space 63
Weyl Representation 64
Hida White Noise Analysis 65
White Noise 65
Hida Distributions 66
Lie Algebra of the Hida Rotation Group 68
Two Elemental Stochastic Processes 69
Elements of Operator Algebras 69
KMS States and Tomita-Takesaki Theory 73
Notes 76
Basics of Quantum Mechanics 77
Quantum Dynamics 77
Schrödinger Picture 78
Heisenberg Picture 79
Free Particle 79
Classical Mechanics 79
Quantum Mechanics 80
Scattering 81
Description of General State: Density Operator 81
States and Observables 83
Quantum Parallelism (Duality) 83
States 86
Observables 86
Expectation Values 87
Quantum Oscillator, Creation and Annihilation Operators 88
Symmetries 89
Representations of the Rotation Group: Spin 90
Quantum Probability 91
Conditional Probability and Joint Probability in QP 93
Probability Positive Operator-Valued Measure (POVM) 96
General Frameworks of Quantum Probability 98
Uncertainty Relation 98
Principle of Quantum Measurements 99
Measurement Procedure 100
Reduction of State 101
Composite Systems and Operations 104
Operations and Completely Positive Mappings 105
Measurements as Unitary Transformations on Extended Space 106
POVM and Instruments 107
POVM 107
Instrument 108
Covariant POVM and Instrument 110
Seven Principles of Quantum Mechanics 111
Notes 115
Fundamentals of Classical Information Theory 117
Communication Processes and Channel 117
Entropy 119
Relative Entropy 124
Mutual Entropy 127
Discrete Case 127
Continuous Case 128
Entropy of Information Source 130
Capacity 132
McMillan's Theorem 133
Shannon's Coding Theorem 134
Channel, Transmission Rate and Capacity Rate 135
Coding Theorem 137
Interpretation of Coding Theorem 138
Notes 139
Fundamentals of Quantum Information 140
Communication Processes 141
Quantum Entropy for Density Operators 142
Relative Entropy for Density Operators 146
More About Channels 151
von Neumann Entropy for the Qubit State 157
Quantum Mutual Entropy 158
Generalized (or Quasi) Quantum Mutual Entropy 162
Ergodic Type Theorem 163
Entropy Exchange and Coherent Information 164
Comparison of Various Quantum Mutual Type Entropies 165
Lifting and Beam Splitting 167
Entropies for General Quantum States 173
Sufficiency and Relative Entropy 178
Notes 180
Locality and Entanglement 182
EPR Model and Bohm Model 183
Various Localities 183
Probability and Local Causality 185
Bell's Local Causality 186
EPR Model vs. Bohm and Bell Model 187
Bell's Locality 188
Discussion of Bell's Locality 189
Example of Local Realist Representation for Spins 189
Local Realist Representation for EPR Correlations 190
On Bell's Theorem 193
Bell's Theorem 196
Bell's Theorem and Stochastic Processes 196
CHSH Inequality 197
Various Local Realisms 199
Bell's Local Realism 199
Space and Time in Axioms of Quantum Mechanics 200
Einstein's Local Realism 200
Local Realistic Representation for Quantum Spin Correlations 201
Correlation Functions and Local Realism 201
Entangled States in Space-Time. Disentanglement 203
Local Observations 205
Modified Bell's Equation 205
Separability and Entanglement 208
Entangling Operator 209
True Quantum Entanglement, d- and c-Entanglements 213
Criteria of Entangled States 218
PPT Criterion 218
Degrees of Entanglement 224
Models of Entanglement in Circulant States 230
Computation of DEN 231
Circulant State Model 1 231
Circulant State Model 2 232
Entangled Markov Chains and Their Properties 233
Entangled Quantum Markov Chains 233
Entangled Markov Chains Generated by Unitary Implementable Matrices 237
Notes 240
Quantum Capacity and Coding 241
Channel Capacity 241
Capacity of Quantum Channel 241
Capacity of Classical-Quantum-Classical Channel 243
Bound of Mutual Entropy and Capacity 243
Computation of Capacity 247
Divergence Center 249
Comparison of Capacities 252
Quantum McMillan Type Theorem 255
Entropy Operators in Quantum Systems 255
McMillan's Type Convergence Theorems 257
Coding Type Theorems 260
Notes 264
Information Dynamics and Adaptive Dynamics 265
Complex Systems 265
Information Dynamics 266
State Change and Complexities 268
Adaptive Dynamics 274
Entropy, Information in Classical and Quantum World 276
Schematic Expression of Understanding 276
Adaptive Dynamics-Conceptual Meaning 277
Description of Chaos 277
Chameleon Dynamics 278
Quantum SAT Algorithm 278
Chaos SAT Algorithm 279
Adaptive SAT Algorithm 280
Summary of Adaptive Dynamics 280
A Use of ID: Chaos Degree 281
Algorithm Computing Entropic Chaos Degree (A Use of AD) 283
Logistic Map 284
Lyapunov Exponent lambda( f ) 285
Bernoulli Shift 286
Baker's Transformation 287
Tinkerbell Map 289
ECD with Memory 293
Adaptive Dynamics Describing Chaos 296
Chaos Degree with Adaptivity 298
Time Irreversibility Problem and Functional Mechanics 302
States and Observables in Functional Classical Mechanics 306
Free Motion 309
Newton's Equation for the Average Coordinate 309
Comparison with Quantum Mechanics 310
Liouville Equation and the Newton Equation 311
Corrections to Newton's Equations 313
Time Reversal 315
Reversibility in Classical Mechanics 315
Irreversibility in the Functional Mechanics 316
Mixing and Weak Limit 317
Dynamics of a Particle in a Box 318
Dynamics of a Classical Particle in a Box 318
Dynamics of a Quantum Particle in a Box 319
New Interpretation of Bell's Inequality-Chameleon Dynamics 320
Dynamical Systems: Passive and Adaptive 320
The EPRB-Chameleon Dynamical System 322
Probabilistic Error Model 324
Upper Bounds for CHSH Inequality 325
Notes 326
Mathematical Models of Quantum Computer 327
Quantum Turing Machine 327
Universal Quantum Turing Machine 330
Quantum Gates 333
Quantum Circuits 335
Universal Quantum Gates 337
Problem on the Halting Scheme 338
Destruction of Computation Result 338
QND (Quantum Non-demolition Monitoring)-Type Quantum Turing Machine 339
Problem for Halting 339
Generalized Quantum Turing Machine 341
Notes 343
Quantum Algorithms I 344
Discrete Quantum Fourier Transform and Principle of Quantum Algorithm 344
Hadamard Gate 344
Discrete Quantum Fourier Transformation 344
Principle of Quantum Algorithm 347
Deutsch-Jozsa Algorithm 347
Grover's Search Algorithm 349
Complexity of Grover's Algorithm 356
Accardi-Sabadini's Unitary Operator 363
Notes 367
Quantum Algorithms II 368
Elements of Number Theory 368
Euclid's Algorithm 368
The Euler Function 372
Modular Exponentiation 373
Shor's Quantum Factoring 374
Finding the Order 375
Description of the Algorithm 376
Computational Complexity of Shor's Algorithm 378
Factoring Integers 379
Factoring Algorithm 379
Notes 381
Quantum Algorithm III 382
Quantum Algorithm of SAT Problem 383
Quantum Computing 384
Quantum Algorithm of SAT Problem 387
Example 391
Quantum Chaos Algorithm 393
Amplification Process in SAT Algorithm 394
Computational Complexity of SAT Algorithm 397
Channel Expression of Quantum Algorithm 398
Channel Expression of Unitary Algorithm 398
Channel Expression of Arbitrary Algorithm 399
SAT Algorithm in GQTM 400
SAT Algorithm with Stochastic Limits 405
Notes 408
Quantum Error Correction 409
Three Qubit Code and Fidelity 409
Three Qubit Code 409
Fidelity and Error Correction 411
The Shor Code 414
Calderbank-Shor-Steane Codes 415
General Theory of Quantum Error-Correcting Codes 416
Depolarizing Channel 417
Fault-Tolerant Quantum Computation and the Threshold Theorem 418
Notes 424
Quantum Field Theory, Locality and Entanglement 425
Quantum Electrodynamics (QED) 425
Maxwell Equations 426
Quantization of Electromagnetic Field 427
Casimir Effect 428
Correlation Functions and Photo-Detection 430
Interference: Two-Slits Experiment 431
Lorentz-Invariant Form of Maxwell Equations 432
Dirac Equation 433
Pauli Equation 434
Equations of Quantum Electrodynamics 435
Quantum Fields and Locality 435
Wightman Axioms 435
Algebraic Quantum Theory and Local Channels 436
Quantum Field Theory in Quantum Probability Scheme 437
Expansion of Wave Packet 440
Relativistic Particles 440
Space Dependence of the Dirac Correlation Function 441
Wave Packets 444
Calculations and Results 446
Results 447
Computation of gij(0) 448
Computation of gij(1) 448
Noncommutative Spectral Theory and Local Realism 449
Contextual Approach 451
Contextual Classical and Quantum Probability 451
Notes 452
Quantum Cryptography 453
Private Key Cryptosystems 454
Julius Caesar's Cryptosystem 454
Symmetric Cryptosystems-DES and GOST 455
Public Key Cryptography and RSA Cryptosystem 455
The RSA Protocol 456
Mathematical Basis of the RSA Protocol 457
Entropic Uncertainty Relations 458
No-cloning Theorem 459
The BB84 Quantum Cryptographic Protocol 461
The BB84 Protocol 461
The Protocol 462
The Security of BB84 462
Ultimate Security Proofs 463
The EPRBE Quantum Cryptographic Protocol 464
Quantum Nonlocality and Cryptography 464
Bell's Inequality and Localized Detectors 465
The EPRBE Quantum Key Distribution 466
Gaussian Wave Functions 467
Notes 469
Quantum Teleportation 470
Channel Expression of Quantum Teleportation 471
BBCJPW Model of Teleportation 473
Weak Teleportation and Uniqueness of Key 474
Perfect Teleportation in Bose Fock Space 477
Basic Notations and Facts 478
Symmetric (Bose) Fock Space 478
Fichtner-Freudenberg Expression of Fock Space 479
Basic Facts in Symmetric Fock Space 480
A Perfect Teleportation 484
Non-perfect Teleportation in Bose Fock Space 486
Fidelity in Teleportation 492
Spatially Separated Teleportation 497
Teleportation of States Inside Rk 497
Alice and Bob Are Spatially Separated 500
Model of Continuous Teleportation 502
Scheme of Continuous Variable Teleportation 502
Entangled State Employed by Continuous Teleportation 505
Quantum Teleportation with Non-maximally Entangled States 506
Basic Setting 507
New Scheme of Teleportation 509
Examples 512
Perfect Teleportation for Non-maximally Entangled State 513
Notes 515
Physical Nanosystems 516
Quantum Dots 516
Quantum Wells, Quantum Wires, Quantum Dots 516
Fock-Darwin Hamiltonian and Interacting Electrons in Quantum Dot 517
Properties of Quantum Dots 519
Quantum Computation with Quantum Dots 519
Quantum Communication Experiments 520
Quantum Cryptography and Teleportation 520
Experimental Realizations of Quantum Computations 521
Ion Traps 521
Nuclear Magnetic Resonance 523
Atomic Quantum Computer 525
Parametric Down Conversion 527
Notes 530
Miscellaneous in Quantum Theory and Information 531
Lifting and Conditional Expectation 531
Lifting 531
Conditional Probability 532
Various Liftings 535
Various Topics on Entropies 538
Boltzmann's Entropy 538
Entropy Increase 538
Gibbs Entropy 539
Boltzmann Entropy in Quantum Mechanics 540
Thermodynamic Quantities 540
von Neumann Microscopic and Macroscopic Entropies 541
Entropy Change for Open Systems 541
Reversible and Irreversible Processes 544
Entropy Production 545
Quantum Dynamical Entropy 549
Formulation by CNT 549
Kolmogorov-Sinai Type Complexities 549
Model Computation 551
Various Quantum Dynamical Entropies 553
Some Models 563
Fractal Dimension of State 568
Applications 570
Entanglement in Janes-Cummings Model 572
Decoherence in Quantum Systems: Stochastic Limit 575
Reducing of Decoherence in Quantum Computers 576
Stochastic Limit 577
Properties of the Quantum Baker's Map 581
Quantum vs. Classical Dynamics 583
Coherent States for the Quantum Baker's Map 585
Expectation Values and Chaos Degree 588
Numerical Simulation of the Chaos Degree and Classical-Quantum Correspondence 601
Quantum-Classical Correspondence for Quantum Baker's Map 604
Combined Quantum Baker's Map 608
A Combined Classical Baker's Map and Its Entropic Chaos Degree 608
A Combined Quantum Baker's Map and Its Entropic Chaos Degree 609
Dependence of the Entropic Chaos Degree on the Combination Parameter a 611
Dependence of the Entropic Chaos Degree on N 614
Notes 615
Applications to Life Science 617
Introduction 618
Quantum Theory and Molecular Biology 618
On Quantum Mind and Brain 619
Characteristics of Biological Systems 620
Genome and Proteins 622
Cell 622
Cell Structure 622
Cell Functions 623
Biomolecules 623
Proteins 623
Nucleic Acids (DNA and RNA) 624
Genetic Code 625
Flow of Biological Information 627
Human Genome 627
Sequence Alignment by Quantum Algorithm 628
Outline of Alignment 628
Quantum Algorithm of Multiple Alignment 629
Computational Complexity of Quantum Algorithm 635
A Brain Model Describing the Recognition Process 635
A Quantum-Like Model of Brain 636
Value of Information in Brain 637
Postulates Concerning the Recognition of Signals 638
The Space of Signals 639
The Memory Space 641
The Exchange Operator 642
Processing of the Signals 644
Recognition of Signals 644
Measurements of the EEG-Type 646
Evolution Tree and Study of the HIV and the Influenza A Viruses 646
Genetic Difference and Evolution Tree 647
A Method Constructing Phylogenetic Trees 648
Phylogenetic Tree for Hemoglobin alpha 653
Evolution and Its Tree of HIV by EER 655
Use of the Entropic Chaos Degree to Study Influenza A Viruses and HIV-1 657
Entropic Chaos Degree 661
Evolution Analysis of Influenza A Viruses 662
Results and Discussion 663
Evolution of the HIV-1 Viruses by ECD 668
Code Structure of HIV-1 670
Coding Theory 670
Cyclic Code 671
Convolutional Code 671
Application of Artificial Codes to Genome 672
How to Encode Genes 672
BCH Code 673
Self-orthogonal Code 674
Iwadare Code 675
Code Structure of HIV-1 675
p-adic Modeling of the Genome and the Genetic Code 678
Genetic Code 680
p-adic Genome 682
Some Mathematical Preliminaries and p-adic Codon Space 682
p-adic Genomic and Bioinformation Spaces 684
DNA and RNA Spaces 685
Protein Space 686
p-adic Genetic Code 688
Degeneracy of the Genetic Code 690
Evolution of the Genetic Code 692
Remarks 695
Mathematical Model of Drug Delivery System 697
Mathematical Model 698
Drug Particle Circulation and the Probability of Staying in a Tumor 699
Drug Particle Movement Model 700
Difference Equation for Numerical Calculation 702
EPR Effect of Tumor and Its Mathematical Treatment 702
Results and Discussion 703
An Application of Quantum Information to a Biosystem 707
Folding Problem 707
Molecular Dynamics 708
Quantum Mechanics and Molecular Dynamics 709
Molecular Mechanics 709
Stochastic Dynamics 710
The Brownian Limit 711
Molecular Dynamics for Harmonic Oscillator 711
Examples of Molecular Dynamics Simulations 713
Quantum Algorithm of Protein Folding 714
Quantum Photosynthesis 714
Photosystems 715
Biophysics of Photosynthesis 716
Quantum Mechanics in Photosynthesis 717
Quantum Network Model 719
Chaotic Amplifier and Decreasing of Entropy 720
Channel Representation of Photosynthesis 721
Quantum-Like Models for Cognitive Psychology 722
Quantum-Like Model for Decision-Making in Two-Player Game 722
Pay-off Table of Two-Player Game 722
Decision-Making Process in Player's Mind 723
Definition of Velocities of k and k 724
Decision-Making in PD-Type Game and Irrational Choice 727
Non-Kolmogorovian Structure 728
References 729
Index 760