Boca Raton: Taylor & Francis Group, LLC. – 2008. – 339. In many experimental setups, an electromagnetic ﬁeld interacts with a system, such as an atom, a nucleus or an electron, whose dynamics follows the laws of quantum mechanics. While it is appropriate to treat the latter system as a quantum mechanical one, the electromagnetic ﬁeld can often be treated as a classical ﬁeld, giving predictions that agree with macroscopic observation. This is the so-called semiclassical approximation. This book is an introduction to the analysis and control of quantum dynamics which emphasizes the application of Lie algebra and Lie group theory. It was developed from lecture notes written during a course given at Iowa State University in the spring semester of 2004. This course was intended for advanced undergraduate and beginning graduate students in mathematics, physics and engineering. The book is organized as follows. Chapter 1 introduces the basics of quantum mechanics, with an emphasis on dynamics and a quantum information theory point of view. In Chapter 2, from fundamental physics, a class of models for quantum control systems is derived. In Chapter 3, we study the controllability of quantum systems and, in the process, we introduce many concepts of Lie algebra and Lie group theory as well as Lie transformation groups. Chapter 4 deals with observability of quantum systems and the related problem of quantum state determination and measurement. Chapter 5 introduces Lie group decompositions as a tool to analyze dynamics and design control algorithms. In Chapters 6 and 7 we describe more methods for control. Chapter 8 presents some topics in quantum information theory. Chapter 9 discusses physical set-ups of implementations and applications of quantum control and dynamics Contents Quantum Mechanics Modeling of Quantum Control Systems; Examples Controllability Observability and State Determination Lie Group Decompositions and Control Optimal Control of Quantum Systems More Tools for Quantum Control Analysis of Quantum Evolutions; Entanglement, Entanglement Measures and Dynamics Applications of Quantum Control and Dynamics Positive and Completely Positive Maps, Quantum Operations and Generalized Measurement Theory Lagrangian and Hamiltonian Formalism in Classical Electrodynamics Cartan Semisimplicity Criterion and Calculation of the Levi Decomposition Proof of the Controllability Test of Theorem 3.2.1 The Baker-Campbell-Hausdorﬀ Formula and Some Exponential Formulas Proof of Theorem 6.2.1 References
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