Cover -- Title Page -- Copyright -- Contents -- About the Author -- Preface -- Chapter 1 Introduction -- 1.1 Quantum States -- 1.2 Quantum Systems Control Models -- 1.2.1 Schrödinger Equation -- 1.2.2 Liouville Equation -- 1.2.3 Markovian Master Equations -- 1.2.4 Non-Markovian Master Equations -- 1.3 Structures of Quantum Control Systems -- 1.4 Control Tasks and Objectives -- 1.5 System Characteristics Analyses -- 1.5.1 Controllability -- 1.5.2 Reachability -- 1.5.3 Observability -- 1.5.4 Stability -- 1.5.5 Convergence -- 1.5.6 Robustness -- 1.6 Performance of Control Systems -- 1.6.1 Probability -- 1.6.2 Fidelity -- 1.6.3 Purity -- 1.7 Quantum Systems Control -- 1.7.1 Description of Control Problems -- 1.7.2 Quantum Control Theory and Methods -- 1.8 Overview of the Book -- References -- Chapter 2 State Transfer and Analysis of Quantum Systems on the Bloch Sphere -- 2.1 Analysis of a Two-level Quantum System State -- 2.1.1 Pure State Expression on the Bloch Sphere -- 2.1.2 Mixed States in the Bloch Sphere -- 2.1.3 Control Trajectory on the Bloch Sphere -- 2.2 State Transfer of Quantum Systems on the Bloch Sphere -- 2.2.1 Control of a Single Spin-1/2 Particle -- 2.2.2 Situation with the Minimum Ωt of Control Fields -- 2.2.3 Situation with a Fixed Time T -- 2.2.4 Numerical Simulations and Results Analyses -- References -- Chapter 3 Control Methods of Closed Quantum Systems -- 3.1 Improved Optimal Control Strategies Applied in Quantum Systems -- 3.1.1 Optimal Control of Quantum Systems -- 3.1.2 Improved Quantum Optimal Control Method -- 3.1.3 Krotov-Based Method of Optimal Control -- 3.1.4 Numerical Simulation and Performance Analysis -- 3.2 Control Design of High-Dimensional Spin-1/2 Quantum Systems -- 3.2.1 Coherent Population Transfer Approaches.

3.2.2 Relationships between the Hamiltonian of Spin-1/2 Quantum Systems under Control and the Sequence of Pulses -- 3.2.3 Design of the Control Sequence of Pulses -- 3.2.4 Simulation Experiments of Population Transfer -- 3.3 Comparison of Time Optimal Control for Two-Level Quantum Systems -- 3.3.1 Description of System Model -- 3.3.2 Geometric Control -- 3.3.3 Bang-Bang Control -- 3.3.4 Time Comparisons of Two Control Strategies -- 3.3.5 Numerical Simulation Experiments and Results Analyses -- References -- Chapter 4 Manipulation of Eigenstates-Based on Lyapunov Method -- 4.1 Principle of the Lyapunov Stability Theorem -- 4.2 Quantum Control Strategy Based on State Distance -- 4.2.1 Selection of the Lyapunov Function -- 4.2.2 Design of the Feedback Control Law -- 4.2.3 Analysis and Proof of the Stability -- 4.2.4 Application to a Spin-1/2 Particle System -- 4.3 Optimal Quantum Control Based on the Lyapunov Stability Theorem -- 4.3.1 Description of the System Model -- 4.3.2 Optimal Control Law Design and Property Analysis -- 4.3.3 Simulation Experiments and the Results Comparisons -- 4.4 Realization of the Quantum Hadamard Gate Based on the Lyapunov Method -- 4.4.1 Mathematical Model -- 4.4.2 Realization of the Quantum Hadamard Gate -- 4.4.3 Design of Control Fields -- 4.4.4 Numerical Simulations and Comparison Results Analyses -- References -- Chapter 5 Population Control Based on the Lyapunov Method -- 5.1 Population Control of Equilibrium State -- 5.1.1 Preliminary Notions -- 5.1.2 Control Laws Design -- 5.1.3 Analysis of the Largest Invariant Set -- 5.1.4 Considerations on the Determination of P -- 5.1.5 Illustrative Example -- 5.1.6 Appendix: Proof of Theorem 5.1 -- 5.2 Generalized Control of Quantum Systems in the Frame of Vector Treatment -- 5.2.1 Design of Control Law -- 5.2.2 Convergence Analysis.

5.2.3 Numerical Simulation on a Spin-1/2 System -- 5.3 Population Control of Eigenstates -- 5.3.1 System Model and Control Laws -- 5.3.2 Largest Invariant Set of Control Systems -- 5.3.3 Analysis of the Eigenstate Control -- 5.3.4 Simulation Experiments -- References -- Chapter 6 Quantum General State Control Based on Lyapunov Method -- 6.1 Pure State Manipulation -- 6.1.1 Design of Control Law and Discussion -- 6.1.2 Control System Simulations and Results Analyses -- 6.2 Optimal Control Strategy of the Superposition State -- 6.2.1 Preliminary Knowledge -- 6.2.2 Control Law Design -- 6.2.3 Numerical Simulations -- 6.3 Optimal Control of Mixed-State Quantum Systems -- 6.3.1 Model of the System to be Controlled -- 6.3.2 Control Law Design -- 6.3.3 Numerical Simulations and Results Analyses -- 6.4 Arbitrary Pure State to a Mixed-State Manipulation -- 6.4.1 Transfer from an Arbitrary Pure State to an Eigenstate -- 6.4.2 Transfer from an Eigenstate to a Mixed State by Interaction Control -- 6.4.3 Control Design for a Mixed-State Transfer -- 6.4.4 Numerical Simulation Experiments -- References -- Chapter 7 Convergence Analysis Based on the Lyapunov Stability Theorem -- 7.1 Population Control of Quantum States Based on Invariant Subsets with the Diagonal Lyapunov Function -- 7.1.1 System Model and Control Design -- 7.1.2 Correspondence between any Target Eigenstate and the Value of the Lyapunov Function -- 7.1.3 Invariant Set of Control Systems -- 7.1.4 Numerical Simulations -- 7.1.5 Summary and Discussion -- 7.2 A Convergent Control Strategy of Quantum Systems -- 7.2.1 Problem Description -- 7.2.2 Construction Method of the Observable Operator -- 7.2.3 Proof of Convergence -- 7.2.4 Route Extension Strategy -- 7.2.5 Numerical Simulations.

7.3 Path Programming Control Strategy of Quantum State Transfer -- 7.3.1 Control Law Design Based on the Lyapunov Method in the Interaction Picture -- 7.3.2 Transition Path Programming Control Strategy -- 7.3.3 Numerical Simulations and Results Analyses -- References -- Chapter 8 Control Theory and Methods in Degenerate Cases -- 8.1 Implicit Lyapunov Control of Multi-Control Hamiltonian Systems Based on State Error -- 8.1.1 Control Design -- 8.1.2 Convergence Proof -- 8.1.3 Relation between Two Lyapunov Functions -- 8.1.4 Numerical Simulation and Result Analysis -- 8.2 Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity -- 8.2.1 Control Law Design and Convergence Proof -- 8.2.2 Numerical Simulation and Result Analysis -- 8.3 Implicit Lyapunov Control for the Quantum Liouville Equation -- 8.3.1 Description of Problem -- 8.3.2 Derivation of Control Laws -- 8.3.3 Convergence Analysis -- 8.3.4 Numerical Simulations -- References -- Chapter 9 Manipulation Methods of the General State -- 9.1 Quantum System Schmidt Decomposition and its Geometric Analysis -- 9.1.1 Schmidt Decomposition of Quantum States -- 9.1.2 Definition of Entanglement Degree Based on the Schmidt Decomposition -- 9.1.3 Application of the Schmidt Decomposition -- 9.2 Preparation of Entanglement States in a Two-Spin System -- 9.2.1 Construction of the Two-Spin Systems Model in the Interaction Picture -- 9.2.2 Design of the Control Field Based on the Lyapunov Method -- 9.2.3 Proof of Convergence for the Bell States -- 9.2.4 Numerical Simulations -- 9.3 Purification of the Mixed State for Two-Dimensional Systems -- 9.3.1 Purification by Means of a Probe -- 9.3.2 Purification by Interaction Control -- 9.3.3 Numerical Experiments and Results Comparisons -- 9.3.4 Discussion -- References.

Chapter 10 State Control of Open Quantum Systems -- 10.1 State Transfer of Open Quantum Systems with a Single Control Field -- 10.1.1 Dynamical Model of Open Quantum Systems -- 10.1.2 Derivation of Optimal Control Law -- 10.1.3 Control System Design -- 10.1.4 Numerical Simulations and Results Analyses -- 10.2 Purity and Coherence Compensation through the Interaction between Particles -- 10.2.1 Method of Compensation for Purity and Coherence -- 10.2.2 Analysis of System Evolution -- 10.2.3 Numerical Simulations -- 10.2.4 Discussion -- Appendix 10.A Proof of Equation 10.59 -- References -- Chapter 11 State Estimation, Measurement, and Control of Quantum Systems -- 11.1 State Estimation Methods in Quantum Systems -- 11.1.1 Background of State Estimation of Quantum Systems -- 11.1.2 Quantum State Estimation Methods Based on the Measurement of Identical Copies -- 11.1.3 Quantum State Reconstruction Methods Based on System Theory -- 11.2 Entanglement Detection and Measurement of Quantum Systems -- 11.2.1 Entanglement States -- 11.2.2 Entanglement Witnesses -- 11.2.3 Entanglement Measures -- 11.2.4 Non-linear Separability Criteria -- 11.3 Decoherence Control Based on Weak Measurement -- 11.3.1 Construction of a Weak Measurement Operator -- 11.3.2 Applicability of Weak Measurement -- 11.3.3 Effects on States -- Appendix 11.A Proof of Normed Linear Space(A, ‖‖) -- References -- Chapter 12 State Preservation of Open Quantum Systems -- 12.1 Coherence Preservation in a -Type Three-Level Atom -- 12.1.1 Models and Objectives -- 12.1.2 Design of Control Field -- 12.1.3 Analysis of Singularities Issues -- 12.1.4 Numerical Simulations -- 12.2 Purity Preservation of Quantum Systems by a Resonant Field -- 12.2.1 Problem Description -- 12.2.2 Purity Property Preservation -- 12.2.3 Discussion.

12.3 Coherence Preservation in Markovian Open Quantum Systems.