Control of Quantum Systems – Theory and Methods

Shuang Cong University of Science and Technology of China

About the Author xiii Preface xv 1 Introduction 1 1.1 Quantum States 2 1.2 Quantum Systems Control Models 3 1.2.1 Schrodinger 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 t 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 -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 -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;-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 -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