Contents 8
Preface 6
Coherent Quantum Control of A-Atoms through the Stochastic Limit 14
1 Introduction 14
2 An atom in a laser field 16
3 The stochastic limit 18
4 The quantum master equation 22
5 Stationary states for a two-level atom 23
6 Stationary states for a three-level atom 25
7 Three-level lambda-atom 27
8 Two-level 3-times degenerate atom 28
9 Conclusions 29
References 30
Recent Advances in Quantum White Noise Calculus 31
1 Emergence of white noise equations from classical quantum mechanics 31
2 Quantum white noise unitary evolutions 33
3 Higher powers of white noise 35
4 Appplications to quantum stochastic control 36
References 39
Control of Quantum States by Decoherence 41
1 Introduction 41
2 A master equation driving to a pre-assigned state 43
3 A microscopic model 44
4 The Master equation 45
5 Assimilation of the state 48
6 Purification 52
7 Example: Entangled state with Spin-Boson model 53
References 58
Logical Operations Realized on the Ising Chain of N Qubits 59
Joint Extension of States of Fermion Subsystems 65
1 Introduction 65
2 Notation for Fermion Systems 66
3 Product Extension 66
4 A Pair of Pure and General States for Two Subsystems 68
5 Examples 70
References 71
Quantum Filtering and Optimal Feedback Control of a Gaussian Quantum Free Particle 72
1 Inroduction 72
2 The Model 73
3 Quantum Filtering 75
4 Control 78
5 Discussion 81
References 81
On Existence of Quantum Zeno Dynamics 85
1 Introduction and the result 85
2 Proof sketch of Theorems 1.1 and 1.2 89
3 An example 92
References 93
Invariant Subspaces and Control of Decoherence 94
1 Introduction 94
2 Generalities and notation 95
3 Control procedures 98
4 The Zeno subspaces 105
5 Summary and concluding remarks 106
References 108
Clauser-Horne Inequality for Electron Counting Statistics in Multiterminal Mesoscopic Conductors 110
1 Introduction 110
2 CH inequality for the full counting statistics 112
3 Results 114
4 Conclusions 124
References 124
Fidelity of Quantum Teleportation Model Using Beam Splittings 126
1 Introduction 126
2 Basic Notions and Notations 127
3 Teleportation scheme with tests 4 134
4 Teleportation and fidelity 135
References 142
Quantum Logical Gates Realized by Beam Splittings 144
1 Introduction 145
2 The Boson Fock Space 147
3 A General Beam Splitting Model 150
4 The Splitting Procedures 155
5 Splitting Functions fulfilling the Truth Table 156
References 159
Information Divergence for Quantum Channels 162
1 Introduction 162
2 Axiomatic Properties of Relative Information 163
3 Examples of Relative Information for States 165
4 Relative Information for Channels 169
References 176
On the Uniqueness Theorem in Quantum Information Geometry 180
0 Introduction 180
1 Extended Chentsov Theorem for noncommutative (finite-dimensional matrix) manifolds 182
2 Derivation and Frechet differentiation 184
3 Classical and quantum a-divergence 185
4 Proof of the uniqueness for matrix manifolds-Main theorem in the nonparametric version 188
5 Parametrization of the matrix manifold 191
6 A Concluding Remark-on Tsallis statistics 192
References 193
Noncanonical Representations of a Multi-dimensional Brownian Motion 194
1 Introduction 194
2 d-dimensional Brownian motions 195
References 198
Some of Future Directions of White Noise Theory 199
1 Introduction 199
2 White noise analysis 199
3 New directions. (Our plans.) 201
4 Innovation 201
References 207
Information Innovation and Elemental Random Field 208
1 Introduction 208
2 White noise analysis 208
3 Our plans 210
4 Innovation 210
5 Path integrals 212
6 Concluding remark 215
References 215
Generalized Quantum Turing Machine and Its Application to the SAT Chaos Algorithm 217
1 Introduction 217
2 Generalized Quantum Turing Machine 218
3 SAT Problem 223
4 SAT Algorithm 226
5 SAT algorithm in GQTM 231
References 237
A Stroboscopic Approach to Quantum Tomography 239
1 Introduction 239
2 Polynomial Representation of Flows 242
3 The Minimal Number of Observables for Qudits Governed by Gaussian Semigroups 244
References 247
Positive Maps and Separable States in Matrix Algebras 248
1 Introduction 248
2 A Class of Positive Maps in Mn 250
References 256
Simulating Open Quantum Systems with Trapped Ions 258
1 Introduction 258
2 Quantum Brownian motion: the exact dynamics 260
3 Time evolution of the mean energy: Lindblad-type and non-Lindblad-type dynamics 263
4 Experiment for simulating QBM with trapped ions 266
5 Conclusions 269
References 269
A Purification Scheme and Entanglement Distillations 272
1 Introduction 272
2 General Framework of Purification Through Repeated Measurements 273
3 Purification of a Qubit 276
4 Entanglement Distillation I 278
5 Entanglement Distillation II 281
6 Summary 284
References 285
Generalized Sectors and Adjunctions to Control Micro-Macro Transitions 287
1 Unified scheme for micro and macro 287
2 Broken symmetry in micro-macro composite system; B=order parameter of broken scale inv. 291
3 Summary and outlook: method of \ 296
References 297
Saturation of an Entropy Bound and Quantum Markov States 298
1 Introduction 298
2 Strong subadditivity of entropy 299
3 Quantum Markov states 301
References 303
An Infinite Dimensional Laplacian Acting on Some Class of Levy White Noise Functionals 305
1 Introduction 305
2 Basic Elememts of White Noise Calculus 306
3 The Levy Laplacian Acting on Levy White Noise Functionals 308
4 The Levy Laplacian Acting on WNF-valued Functions 310
5 An Infinite Dimensional Stochastic Process Associated with the Levy Laplacian 311
References 314
Structure of Linear Processes 317
1 Introduction 317
2 An illustrative example Linear functionals of Poisson noise 318
3 Innovation of a linear process 319
4 Determination of kernels 321
References 324
Group Theory of Dynamical Maps 326
On Quantum Analysis Quantum Transfer-Matrix Method and Effective Information Entropy 337
1 Introduction 337
2 Quantum Analysis 337
3 Quantum Transfer-Matrix Method based on Exponential Product Formulas and its Connection with Quantum Information 339
4 Effective Information Entropy 341
5 Summary and Discussion 342
References 343
Nonequilibrium Steady States for a Harmonic Oscillator Interacting with Two Bose Fields - Stochastic Limit Approach and C* Algebraic Approach 345
1 Introduction 345
2 Model 346
3 Stochastic Limit Approach 348
4 Nonequilibrium Steady States in C*-algebraic Approach 349
5 Weak coupling limits 355
6 Summary and Discussion 356
References 363
Control of Decoherence with Multipulse Application 365
1 Introduction 365
2 Formulation 366
3 Numerical evaluation 368
4 Discussion and Concluding Remarks 370
References 372
Quantum Entanglement Purification and Linear-optics Quantum Gates with Photonic Qubits 373
1 Entanglement 373
2 Quantum Key Distribution using Polarization Entangled Photons 376
3 Purification of Entanglement 379
References 382
On Quantum Mutual Type Measures and Capacity 383
1 Quantum Channels 384
2 Ohya Mutual Entropy 388
3 Comparison of various quantum mutual type entropies 390
4 Quantum Capacity 392
References 395