Control of Quantum Systems: Theory and Methods

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 ????-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
		12.3.1 Problem Formulation
		12.3.2 Design of Control Variables
		12.3.3 Numerical Simulations
		12.3.4 Discussion
	Appendix 12.A Derivation of HC
	References
13 State Manipulation in Decoherence-Free Subspace
	13.1 State Transfer and Coherence Maintainance Based on DFS for a Four-Level Energy Open Quantum System
		13.1.1 Construction of DFS and the Desired Target State
		13.1.2 Design of the Lyapunov-Based Control Law for State Transfer
		13.1.3 Numerical Simulations
	13.2 State Transfer Based on a Decoherence-Free Target State for a ????-Type N-Level Atomic System
		13.2.1 Construction of the Decoherence-Free Target State
		13.2.2 Design of the Lyapunov-Based Control Law for State Transfer
		13.2.3 Numerical Simulations and Results Analyses
	13.3 Control of Quantum States Based on the Lyapunov Method in Decoherence-Free Subspaces
		13.3.1 Problem Description
		13.3.2 Control Design in the Interaction Picture
		13.3.3 Construction of P and Convergence Analysis
		13.3.4 Numerical Simulation Examples and Discussion
	References
14 Dynamic Decoupling Quantum Control Methods
	14.1 Phase Decoherence Suppression of an n-Level Atom in Ξ-Configuration with Bang-Bang Controls
		14.1.1 Dynamical Decoupling Mechanism
		14.1.2 Design of the Bang–Bang Operations Group in Phase Decoherence
		14.1.3 Examples of Design
	14.2 Optimized Dynamical Decoupling in Ξ-Type n-Level Atom
		14.2.1 Periodic Dynamical Decoupling
		14.2.2 Uhrig Dynamical Decoupling
		14.2.3 Behaviors of Quantum Coherence under Various Dynamical Decoupling Schemes
		14.2.4 Examples
		14.2.5 Discussion
	14.3 An Optimized Dynamical Decoupling Strategy to Suppress Decoherence
		14.3.1 Universal Dynamical Decoupling for a Qubit
		14.3.2 An Optimized Dynamical Decoupling Scheme
		14.3.3 Simulation and Comparison
		14.3.4 Discussion
	References
15 Trajectory Tracking of Quantum Systems
	15.1 Orbit Tracking of Quantum States Based on the Lyapunov Method
		15.1.1 Description of the System Model
		15.1.2 Design of Control Law
		15.1.3 Numerical Simulation Experiments and Results Analysis
	15.2 Orbit Tracking Control of Quantum Systems
		15.2.1 System Model and Control Law Design
		15.2.2 Numerical Simulation Experiments
	15.3 Adaptive Trajectory Tracking of Quantum Systems
		15.3.1 Description of the System Model
		15.3.2 Control System Design and Characteristic Analysis
		15.3.3 Numerical Simulation and Result Analysis
	15.4 Convergence of Orbit Tracking for Quantum Systems
		15.4.1 Description of the Control System Model
		15.4.2 Control Law Derivation
		15.4.3 Convergence Analysis
		15.4.4 Applications and Experimental Results Analyses
	References
Index