Preface xv

1 Introduction 1

1.1 Quantum States 2

1.2 Quantum Systems Control Models 3

1.2.1 Schrödinger Equation 4

1.2.2 Liouville Equation 4

1.2.3 Markovian Master Equations 5

1.2.4 Non-Markovian Master Equations 5

1.3 Structures of Quantum Control Systems 6

1.4 Control Tasks and Objectives 8

1.5 System Characteristics Analyses 9

1.5.1 Controllability 9

1.5.2 Reachability 9

1.5.3 Observability 10

1.5.4 Stability 10

1.5.5 Convergence 10

1.5.6 Robustness 10

1.6 Performance of Control Systems 11

1.6.1 Probability 11

1.6.2 Fidelity 11

1.6.3 Purity 12

1.7 Quantum Systems Control 13

1.7.1 Description of Control Problems 13

1.7.2 Quantum Control Theory and Methods 13

1.8 Overview of the Book 16

References 18

2 State Transfer and Analysis of Quantum Systems on the Bloch Sphere 21

2.1 Analysis of a Two-level Quantum System State 21

2.1.1 Pure State Expression on the Bloch Sphere 21

2.1.2 Mixed States in the Bloch Sphere 24

2.1.3 Control Trajectory on the Bloch Sphere 26

2.2 State Transfer of Quantum Systems on the Bloch Sphere 27

2.2.1 Control of a Single Spin-1/2 Particle 28

2.2.2 Situation with the Minimum Ωt of Control Fields 30

2.2.3 Situation with a Fixed Time T 31

2.2.4 Numerical Simulations and Results Analyses 33

References 37

3 Control Methods of Closed Quantum Systems 39

3.1 Improved Optimal Control Strategies Applied in Quantum Systems 39

3.1.1 Optimal Control of Quantum Systems 40

3.1.2 Improved Quantum Optimal Control Method 42

3.1.3 Krotov-Based Method of Optimal Control 43

3.1.4 Numerical Simulation and Performance Analysis 45

3.2 Control Design of High-Dimensional Spin-1/2 Quantum Systems 48

3.2.1 Coherent Population Transfer Approaches 48

3.2.2 Relationships between the Hamiltonian of Spin-1/2 Quantum Systems under Control and the Sequence of Pulses 49

3.2.3 Design of the Control Sequence of Pulses 52

3.2.4 Simulation Experiments of Population Transfer 53

3.3 Comparison of Time Optimal Control for Two-Level Quantum Systems 57

3.3.1 Description of System Model 58

3.3.2 Geometric Control 59

3.3.3 Bang-Bang Control 61

3.3.4 Time Comparisons of Two Control Strategies 64

3.3.5 Numerical Simulation Experiments and Results Analyses 66

References 71

4 Manipulation of Eigenstates – Based on Lyapunov Method 73

4.1 Principle of the Lyapunov Stability Theorem 74

4.2 Quantum Control Strategy Based on State Distance 75

4.2.1 Selection of the Lyapunov Function 76

4.2.2 Design of the Feedback Control Law 77

4.2.3 Analysis and Proof of the Stability 78

4.2.4 Application to a Spin-1/2 Particle System 80

4.3 Optimal Quantum Control Based on the Lyapunov Stability Theorem 81

4.3.1 Description of the System Model 82

4.3.2 Optimal Control Law Design and Property Analysis 84

4.3.3 Simulation Experiments and the Results Comparisons 86

4.4 Realization of the Quantum Hadamard Gate Based on the Lyapunov Method 88

4.4.1 Mathematical Model 89

4.4.2 Realization of the Quantum Hadamard Gate 90

4.4.3 Design of Control Fields 92

4.4.4 Numerical Simulations and Comparison Results Analyses 94

References 96

5 Population Control Based on the Lyapunov Method 99

5.1 Population Control of Equilibrium State 99

5.1.1 Preliminary Notions 99

5.1.2 Control Laws Design 100

5.1.3 Analysis of the Largest Invariant Set 101

5.1.4 Considerations on the Determination of P 104

5.1.5 Illustrative Example 105

5.1.6 Appendix: Proof of Theorem 5.1 107

5.2 Generalized Control of Quantum Systems in the Frame of Vector Treatment 110

5.2.1 Design of Control Law 110

5.2.2 Convergence Analysis 113

5.2.3 Numerical Simulation on a Spin-1/2 System 114

5.3 Population Control of Eigenstates 117

5.3.1 System Model and Control Laws 117

5.3.2 Largest Invariant Set of Control Systems 118

5.3.3 Analysis of the Eigenstate Control 118

5.3.4 Simulation Experiments 119

References 123

6 Quantum General State Control Based on Lyapunov Method 125

6.1 Pure State Manipulation 125

6.1.1 Design of Control Law and Discussion 125

6.1.2 Control System Simulations and Results Analyses 129

6.2 Optimal Control Strategy of the Superposition State 131

6.2.1 Preliminary Knowledge 132

6.2.2 Control Law Design 133

6.2.3 Numerical Simulations 134

6.3 Optimal Control of Mixed-State Quantum Systems 135

6.3.1 Model of the System to be Controlled 136

6.3.2 Control Law Design 137

6.3.3 Numerical Simulations and Results Analyses 142

6.4 Arbitrary Pure State to a Mixed-State Manipulation 145

6.4.1 Transfer from an Arbitrary Pure State to an Eigenstate 146

6.4.2 Transfer from an Eigenstate to a Mixed State by Interaction Control 147

6.4.3 Control Design for a Mixed-State Transfer 149

6.4.4 Numerical Simulation Experiments 151

References 154

7 Convergence Analysis Based on the Lyapunov Stability Theorem 155

7.1 Population Control of Quantum States Based on Invariant Subsets with the Diagonal Lyapunov Function 155

7.1.1 System Model and Control Design 155

7.1.2 Correspondence between any Target Eigenstate and the Value of the Lyapunov Function 156

7.1.3 Invariant Set of Control Systems 157

7.1.4 Numerical Simulations 161

7.1.5 Summary and Discussion 164

7.2 A Convergent Control Strategy of Quantum Systems 165

7.2.1 Problem Description 165

7.2.2 Construction Method of the Observable Operator 166

7.2.3 Proof of Convergence 168

7.2.4 Route Extension Strategy 173

7.2.5 Numerical Simulations 174

7.3 Path Programming Control Strategy of Quantum State Transfer 176

7.3.1 Control Law Design Based on the Lyapunov Method in the Interaction Picture 177

7.3.2 Transition Path Programming Control Strategy 178

7.3.3 Numerical Simulations and Results Analyses 182

References 186

8 Control Theory and Methods in Degenerate Cases 187

8.1 Implicit Lyapunov Control of Multi-Control Hamiltonian Systems Based on State Error 187

8.1.1 Control Design 188

8.1.2 Convergence Proof 192

8.1.3 Relation between Two Lyapunov Functions 193

8.1.4 Numerical Simulation and Result Analysis 193

8.2 Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity 195

8.2.1 Control Law Design and Convergence Proof 195

8.2.2 Numerical Simulation and Result Analysis 199

8.3 Implicit Lyapunov Control for the Quantum Liouville Equation 200

8.3.1 Description of Problem 201

8.3.2 Derivation of Control Laws 202

8.3.3 Convergence Analysis 205

8.3.4 Numerical Simulations 209

References 211

9 Manipulation Methods of the General State 213

9.1 Quantum System Schmidt Decomposition and its Geometric Analysis 213

9.1.1 Schmidt Decomposition of Quantum States 214

9.1.2 Definition of Entanglement Degree Based on the Schmidt Decomposition 215

9.1.3 Application of the Schmidt Decomposition 216

9.2 Preparation of Entanglement States in a Two-Spin System 220

9.2.1 Construction of the Two-Spin Systems Model in the Interaction Picture 220

9.2.2 Design of the Control Field Based on the Lyapunov Method 223

9.2.3 Proof of Convergence for the Bell States 226

9.2.4 Numerical Simulations 227

9.3 Purification of the Mixed State for Two-Dimensional Systems 230

9.3.1 Purification by Means of a Probe 230

9.3.2 Purification by Interaction Control 232

9.3.3 Numerical Experiments and Results Comparisons 233

9.3.4 Discussion 234

References 235

10 State Control of Open Quantum Systems 237

10.1 State Transfer of Open Quantum Systems with a Single Control Field 237

10.1.1 Dynamical Model of Open Quantum Systems 237

10.1.2 Derivation of Optimal Control Law 238

10.1.3 Control System Design 241

10.1.4 Numerical Simulations and Results Analyses 242

10.2 Purity and Coherence Compensation through the Interaction between Particles 246

10.2.1 Method of Compensation for Purity and Coherence 247

10.2.2 Analysis of System Evolution 250

10.2.3 Numerical Simulations 253

10.2.4 Discussion 255

Appendix 10.A Proof of Equation 10.59 257

References 258

11 State Estimation, Measurement, and Control of Quantum Systems 261

11.1 State Estimation Methods in Quantum Systems 261

11.1.1 Background of State Estimation of Quantum Systems 262

11.1.2 Quantum State Estimation Methods Based on the Measurement of Identical Copies 262

11.1.3 Quantum State Reconstruction Methods Based on System Theory 267

11.2 Entanglement Detection and Measurement of Quantum Systems 268

11.2.1 Entanglement States 269

11.2.2 Entanglement Witnesses 271

11.2.3 Entanglement Measures 273

11.2.4 Non-linear Separability Criteria 277

11.3 Decoherence Control Based on Weak Measurement 278

11.3.1 Construction of a Weak Measurement Operator 279

11.3.2 Applicability of Weak Measurement 280

11.3.3 Effects on States 282

Appendix 11.A Proof of Normed Linear Space (A, ¡¬ • ¡¬) 286

References 287

12 State Preservation of Open Quantum Systems 291

12.1 Coherence Preservation in a Λ-Type Three-Level Atom 291

12.1.1 Models and Objectives 292

12.1.2 Design of Control Field 294

12.1.3 Analysis of Singularities Issues 297

12.1.4 Numerical Simulations 299

12.2 Purity Preservation of Quantum Systems by a Resonant Field 301

12.2.1 Problem Description 302

12.2.2 Purity Property Preservation 303

12.2.3 Discussion 306

12.3 Coherence Preservation in Markovian Open Quantum Systems 307

12.3.1 Problem Formulation 308

12.3.2 Design of Control Variables 311

12.3.3 Numerical Simulations 313

12.3.4 Discussion 315

Appendix 12.A Derivation of HC 316

References 317

13 State Manipulation in Decoherence-Free Subspace 321

13.1 State Transfer and Coherence Maintainance Based on DFS for a Four-Level Energy Open Quantum System 321

13.1.1 Construction of DFS and the Desired Target State 322

13.1.2 Design of the Lyapunov-Based Control Law for State Transfer 325

13.1.3 Numerical Simulations 326

13.2 State Transfer Based on a Decoherence-Free Target State for a Λ-Type N-Level Atomic System 328

13.2.1 Construction of the Decoherence-Free Target State 328

13.2.2 Design of the Lyapunov-Based Control Law for State Transfer 331

13.2.3 Numerical Simulations and Results Analyses 332

13.3 Control of Quantum States Based on the Lyapunov Method in Decoherence-Free Subspaces 336

13.3.1 Problem Description 336

13.3.2 Control Design in the Interaction Picture 338

13.3.3 Construction of P and Convergence Analysis 339

13.3.4 Numerical Simulation Examples and Discussion 345

References 348

14 Dynamic Decoupling Quantum Control Methods 351

14.1 Phase Decoherence Suppression of an n-Level Atom in Ξ;-Configuration with Bang-Bang Controls 351

14.1.1 Dynamical Decoupling Mechanism 352

14.1.2 Design of the Bang–Bang Operations Group in Phase Decoherence 355

14.1.3 Examples of Design 357

14.2 Optimized Dynamical Decoupling in Ξ-Type n-Level Atom 360

14.2.1 Periodic Dynamical Decoupling 361

14.2.2 Uhrig Dynamical Decoupling 361

14.2.3 Behaviors of Quantum Coherence under Various Dynamical Decoupling Schemes 362

14.2.4 Examples 365

14.2.5 Discussion 366

14.3 An Optimized Dynamical Decoupling Strategy to Suppress Decoherence 366

14.3.1 Universal Dynamical Decoupling for a Qubit 367

14.3.2 An Optimized Dynamical Decoupling Scheme 369

14.3.3 Simulation and Comparison 369

14.3.4 Discussion 375

References 378

15 Trajectory Tracking of Quantum Systems 381

15.1 Orbit Tracking of Quantum States Based on the Lyapunov Method 382

15.1.1 Description of the System Model 382

15.1.2 Design of Control Law 384

15.1.3 Numerical Simulation Experiments and Results Analysis 385

15.2 Orbit Tracking Control of Quantum Systems 389

15.2.1 System Model and Control Law Design 390

15.2.2 Numerical Simulation Experiments 391

15.3 Adaptive Trajectory Tracking of Quantum Systems 394

15.3.1 Description of the System Model 396

15.3.2 Control System Design and Characteristic Analysis 398

15.3.3 Numerical Simulation and Result Analysis 400

15.4 Convergence of Orbit Tracking for Quantum Systems 402

15.4.1 Description of the Control System Model 403

15.4.2 Control Law Derivation 404

15.4.3 Convergence Analysis 404

15.4.4 Applications and Experimental Results Analyses 411

References 416

Index 419

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