Introduction to Quantum Control and Dynamics, 2nd Edition
- Length: 416 pages
- Edition: 2
- Language: English
- Publisher: Chapman and Hall/CRC
- Publication Date: 2021-07-19
- ISBN-10: 0367507900
- ISBN-13: 9780367507909
- Sales Rank: #7595444 (See Top 100 Books)
The introduction of control theory in quantum mechanics has created a rich, new interdisciplinary scientific field, which is producing novel insight into important theoretical questions at the heart of quantum physics. Exploring this emerging subject, Introduction to Quantum Control and Dynamics presents the mathematical concepts and fundamental physics behind the analysis and control of quantum dynamics, emphasizing the application of Lie algebra and Lie group theory.
To advantage students, instructors and practitioners, and since the field is highly interdisciplinary, this book presents an introduction with all the basic notions in the same place. The field has seen a large development in parallel with the neighboring fields of quantum information, computation and communication. The author has maintained an introductory level to encourage course use.
After introducing the basics of quantum mechanics, the book derives a class of models for quantum control systems from fundamental physics. It examines the controllability and observability of quantum systems and the related problem of quantum state determination and measurement. The author also uses Lie group decompositions as tools to analyze dynamics and to design control algorithms. In addition, he describes various other control methods and discusses topics in quantum information theory that include entanglement and entanglement dynamics.
Changes to the New Edition:
- New Chapter 4: Uncontrollable Systems and Dynamical Decomposition
- New section on quantum control landscapes
- A brief discussion of the experiments that earned the 2012 Nobel Prize in Physics
- Corrections and revised concepts are made to improve accuracy
Armed with the basics of quantum control and dynamics, readers will invariably use this interdisciplinary knowledge in their mathematics, physics and engineering work.
Cover Half Title Series Page Title Page Copyright Page Contents Preface to the First Edition Preface to the Second Edition 1. Quantum Mechanics 1.1. States and Operators 1.1.1. State of a quantum system 1.1.2. Linear operators 1.1.3. State of composite systems and tensor product spaces 1.1.4. State of an ensemble; density operator 1.1.5. Vector and matrix representation of states and operators 1.2. Observables and Measurement 1.2.1. Observables 1.2.2. The measurement postulate 1.2.3. Measurements on ensembles 1.3. Dynamics of Quantum Systems 1.3.1. Schrodinger picture 1.3.2. Heisenberg and interaction pictures 1.4. Notes and References 1.4.1. Interpretation of quantum dynamics as information processing 1.4.2. Direct sum versus tensor product for composite systems 1.5. Exercises 2. Modeling of Quantum Control Systems; Examples 2.1. Classical Theory of Interaction of Particles and Fields 2.1.1. Classical electrodynamics 2.2. Quantum Theory of Interaction of Particles and Fields 2.2.1. Canonical quantization 2.2.2. Quantum mechanical Hamiltonian 2.3. Introduction of Approximations, Modeling and Applications to Molecular Systems 2.3.1. Approximations for molecular and atomic systems 2.3.2. Controlled Schrodinger wave equation 2.4. Spin Dynamics and Control 2.4.1. Introduction of the spin degree of freedom in the dynamics of matter and fields 2.4.2. Spin networks as control systems 2.5. Mathematical Structure of Quantum Control Systems 2.5.1. Control of ensembles 2.5.2. Control of the evolution operator 2.5.3. Output of a quantum control system 2.6. Notes and References 2.6.1. An example of canonical quantization: The quantum harmonic oscillator 2.6.2. On the models introduced for quantum control systems 2.7. Exercises 3. Controllability 3.1. Lie Algebras and Lie Groups 3.1.1. Basic definitions for Lie algebras 3.1.2. Lie groups 3.2. Controllability Test: The Dynamical Lie Algebra 3.2.1. On the proof of the controllability test 3.2.2. Procedure to generate a basis of the dynamical Lie algebra 3.2.3. Uniform nite generation of compact Lie groups 3.2.4. Controllability as a generic property 3.2.5. Reachable set from some time onward 3.3. Notions of Controllability for the State 3.3.1. Pure state controllability 3.3.2. Test for pure state controllability 3.3.3. Equivalent state controllability 3.3.4. Equality of orbits and practical tests 3.3.5. Density matrix controllability 3.4. Notes and References 3.4.1. Alternate tests of controllability 3.4.2. Pure state controllability and existence of constants of motion 3.4.3. Bibliographical notes 3.4.4. Some open problems 3.5. Exercises 4. Uncontrollable Systems and Dynamical Decomposition 4.1. Dynamical Decomposition Starting from a Basis of the Dynamical Lie Algebra 4.1.1. Finding the simple ideals 4.1.2. Decomposition of the dynamics 4.2. Tensor Product Structure of the Dynamical Lie Algebra 4.2.1. Some representation theory and the Schur Lemma 4.2.2. Tensor product structure for the irreducible representation of the product of two groups 4.2.3. Tensor product structure of the dynamical Lie algebra 4.3. Dynamical Decomposition Starting from a Group of Symmetries; Subspace Controllability 4.3.1. Some more representation theory of nite groups: Group algebra, regular representation and Young Symmetrizers 4.3.2. Structure of the representations of uG(n) and G 4.3.3. Subspace controllability 4.3.4. Decomposition without knowing the generalized Young Symmetrizers 4.4. Notes and References 4.5. Exercises 5. Observability and State Determination 5.1. Quantum State Tomography 5.1.1. Example: Quantum tomography of a spin-1/2 particle 5.1.2. General quantum tomography 5.1.3. Example: Quantum tomography of a spin-1/2 particle (ctd.) 5.2. Observability 5.2.1. Equivalence classes of indistinguishable states; partition of the state space 5.3. Observability and Methods for State Reconstruction 5.3.1. Observability conditions and tomographic methods 5.3.2. System theoretic methods for quantum state reconstruction 5.4. Notes and References 5.5. Exercises 6. Lie Group Decompositions and Control 6.1. Decompositions of SU(2) and Control of Two-Level Systems 6.1.1. The Lie groups SU(2) and SO(3) 6.1.2. Euler decomposition of SU(2) and SO(3) 6.1.3. Determination of the angles in the Euler decomposition of SU(2) 6.1.4. Application to the control of two-level quantum systems 6.2. Decomposition in Planar Rotations 6.3. Cartan Decompositions 6.3.1. Cartan decomposition of semisimple Lie algebras 6.3.2. The decomposition theorem for Lie groups 6.3.3. Refinement of the decomposition; Cartan subalgebras 6.3.4. Cartan decompositions of su(n) 6.3.5. Cartan involutions of su(n) and quantum symmetries 6.3.6. Computation of the factors in the Cartan decompositions of SU(n) 6.4. Examples of Application of Decompositions to Control 6.4.1. Control of two coupled spin-1/2 particles with Ising interaction 6.4.2. Control of two coupled spin-1/2 particles with Heisenberg interaction 6.5. Notes and References 6.6. Exercises 7. Optimal Control of Quantum Systems 7.1. Formulation of the Optimal Control Problem 7.1.1. Optimal control problems of Mayer, Lagrange and Bolza 7.1.2. Optimal control problems for quantum systems 7.2. The Necessary Conditions of Optimality 7.2.1. General necessary conditions of optimality 7.2.2. The necessary optimality conditions for quantum control problems 7.3. Example: Optimal Control of a Two-Level Quantum System 7.4. Time Optimal Control of Quantum Systems 7.4.1. The time optimal control problem; bounded control 7.4.2. Minimum time control with unbounded control; Riemannian symmetric spaces 7.5. Numerical Methods for Optimal Control of Quantum Systems 7.5.1. Methods using discretization 7.5.2. Iterative methods 7.5.3. Numerical methods for two points boundary value problems 7.6. Quantum Optimal Control Landscape 7.7. Notes and References 7.8. Exercises 8. More Tools for Quantum Control 8.1. Selective Population Transfer via Frequency Tuning 8.2. Time-Dependent Perturbation Theory 8.3. Adiabatic Control 8.4. STIRAP 8.5. Lyapunov Control of Quantum Systems 8.5.1. Quantum control problems in terms of a Lyapunov function 8.5.2. Determination of the control function 8.5.3. Study of the asymptotic behavior of the state r 8.6. Notes and References 8.7. Exercises 9. Analysis of Quantum Evolutions; Entanglement, Entanglement Measures and Dynamics 9.1. Entanglement of Quantum Systems 9.1.1. Basic definitions and notions 9.1.2. Tests of entanglement 9.1.3. Measures of entanglement and concurrence 9.2. Dynamics of Entanglement 9.2.1. The two qubits example 9.2.2. The odd-even decomposition and concurrence dynamics 9.2.3. Recursive decomposition of dynamics in entangling and local parts 9.3. Local Equivalence of States 9.3.1. General considerations on dimensions 9.3.2. Invariants and polynomial invariants 9.3.3. Some solved cases 9.4. Notes and References 9.5. Exercises 10. Applications of Quantum Control and Dynamics 10.1. Nuclear Magnetic Resonance Experiments 10.1.1. Basics of NMR 10.1.2. Two-dimensional NMR 10.1.3. Control problems in NMR 10.2. Molecular Systems Control 10.2.1. Pulse shaping 10.2.2. Objectives and techniques of molecular control 10.3. Atomic Systems Control; Implementations of Quantum Information Processing with Ion Traps 10.3.1. Physical set-up of the trapped ions quantum information processor 10.3.2. Classical Hamiltonian 10.3.3. Quantum mechanical Hamiltonian 10.3.4. Practical implementation of different interaction Hamiltonians 10.3.5. The control problem: Switching between Hamiltonians 10.4. Notes and References 10.5. Exercises A. Positive and Completely Positive Maps, Quantum Operations and Generalized Measurement Theory A.1. Positive and Completely Positive Maps A.2. Quantum Operations and Operator Sum Representation A.3. Generalized Measurement Theory B. Lagrangian and Hamiltonian Formalism in Classical Electrodynamics B.1. Lagrangian Mechanics B.2. Extension of Lagrangian Mechanics to Systems with Infinite Degrees of Freedom B.3. Lagrangian and Hamiltonian Mechanics for a System of Interacting Particles and Field C. Complements to the Theory of Lie Algebras and Lie Groups C.1. The Adjoint Representation and Killing Form C.2. Cartan Semisimplicity Criterion C.3. Correspondence between Ideals and Normal Sub-groups C.4. Quotient Lie Algebras C.5. Levi Decomposition D. Proof of the Controllability Test of Theorem 3.2.1 E. The Baker-Campbell-Hausdorff Formula and Some Exponential Formulas F. Proof of Theorem 7.2.1 List of Acronyms and Symbols References Index
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