دانلود کتاب Control of Quantum Systems: Theory and Methods

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Control of Quantum Systems: Theory and Methods
Contents
About the Author
Preface
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
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
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
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
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
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
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
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
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
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
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
12 State Preservation of Open Quantum Systems
12.1 Coherence Preservation in a