Classical Topology and Quantum States
Classical Topology and Quantum States

Author: A. P. Balachandran

Publisher: World Scientific

Published: 1991

Total Pages: 386

ISBN-13: 9789810203290

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This book is an introduction to the role of topology in the quantization of classical systems. It is also an introduction to topological solitons with special emphasis on Skyrmions. As regards the first aspect, several issues of current interest are dealt with at a reasonably elementary level. Examples are principal fibre bundles and their role in quantum physics, the possibility of spinorial quantum states in a Lagrangian theory based on tensorial variables, and multiply connected configuration spaces and associated quantum phenomena like the QCD q angle and exotic statistics. The ideas are also illustrated by simple examples such as the spinning particle, the charge-monopole system and strings in 3+1 dimensions. The application of these ideas to quantum gravity is another subject treated at an introductory level. An attempt has been made in this book to introduce the reader to the significance of topology for many distinct physical systems such as spinning particles, the charge- monopole system, strings, Skyrmions, QCD and gravity. The book is an outgrowth of lectures given by the authors at various institutions and conferences.

The Role of Topology in Classical and Quantum Physics
The Role of Topology in Classical and Quantum Physics

Author: Giuseppe Morandi

Publisher: Springer Science & Business Media

Published: 2008-09-11

Total Pages: 254

ISBN-13: 3540466886

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In solid-state physics especially topological techniques have turned out to be extremely useful for modelling and explaining physical properties of matter. This book illustrates various applications of algebraic topology in classical field theory (non-linear sigma-models) and in quantizationsin multiply connected spaces (anyons). It treats Chern-Simon Lagrangians, Berry's phase, the polarization of light and the fractional quantum Hall effect.

Geometry of Quantum States
Geometry of Quantum States

Author: Ingemar Bengtsson

Publisher: Cambridge University Press

Published: 2017-08-18

Total Pages: 637

ISBN-13: 1108293492

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Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen–Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.

Introduction to Topological Quantum Matter & Quantum Computation
Introduction to Topological Quantum Matter & Quantum Computation

Author: Tudor D. Stanescu

Publisher: CRC Press

Published: 2016-12-19

Total Pages: 395

ISBN-13: 1482245949

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What is "topological" about topological quantum states? How many types of topological quantum phases are there? What is a zero-energy Majorana mode, how can it be realized in a solid state system, and how can it be used as a platform for topological quantum computation? What is quantum computation and what makes it different from classical computation? Addressing these and other related questions, Introduction to Topological Quantum Matter & Quantum Computation provides an introduction to and a synthesis of a fascinating and rapidly expanding research field emerging at the crossroads of condensed matter physics, mathematics, and computer science. Providing the big picture, this book is ideal for graduate students and researchers entering this field as it allows for the fruitful transfer of paradigms and ideas amongst different areas, and includes many specific examples to help the reader understand abstract and sometimes challenging concepts. It explores the topological quantum world beyond the well-known topological insulators and superconductors and emphasizes the deep connections with quantum computation. It addresses key principles behind the classification of topological quantum phases and relevant mathematical concepts and discusses models of interacting and noninteracting topological systems, such as the torric code and the p-wave superconductor. The book also covers the basic properties of anyons, and aspects concerning the realization of topological states in solid state structures and cold atom systems. Quantum computation is also presented using a broad perspective, which includes fundamental aspects of quantum mechanics, such as Bell's theorem, basic concepts in the theory of computation, such as computational models and computational complexity, examples of quantum algorithms, and elements of classical and quantum information theory.

Quantum Topology
Quantum Topology

Author: Louis H. Kauffman

Publisher: World Scientific

Published: 1993

Total Pages: 400

ISBN-13: 9789810225759

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This book constitutes a review volume on the relatively new subject of Quantum Topology. Quantum Topology has its inception in the 1984/1985 discoveries of new invariants of knots and links (Jones, Homfly and Kauffman polynomials). These invariants were rapidly connected with quantum groups and methods in statistical mechanics. This was followed by Edward Witten's introduction of methods of quantum field theory into the subject and the formulation by Witten and Michael Atiyah of the concept of topological quantum field theories.This book is a review volume of on-going research activity. The papers derive from talks given at the Special Session on Knot and Topological Quantum Field Theory of the American Mathematical Society held at Dayton, Ohio in the fall of 1992. The book consists of a self-contained article by Kauffman, entitled Introduction to Quantum Topology and eighteen research articles by participants in the special session.This book should provide a useful source of ideas and results for anyone interested in the interface between topology and quantum field theory.

Geometry of Quantum States
Geometry of Quantum States

Author: Ingemar Bengtsson

Publisher: Cambridge University Press

Published: 2007-12-06

Total Pages: 479

ISBN-13: 1139453467

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Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.

Geometric Phases in Classical and Quantum Mechanics
Geometric Phases in Classical and Quantum Mechanics

Author: Dariusz Chruscinski

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 346

ISBN-13: 0817681760

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Several well-established geometric and topological methods are used in this work in an application to a beautiful physical phenomenon known as the geometric phase. This book examines the geometric phase, bringing together different physical phenomena under a unified mathematical scheme. The material is presented so that graduate students and researchers in applied mathematics and physics with an understanding of classical and quantum mechanics can handle the text.

Mathematical Topics Between Classical and Quantum Mechanics
Mathematical Topics Between Classical and Quantum Mechanics

Author: Nicholas P. Landsman

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 547

ISBN-13: 146121680X

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This monograph draws on two traditions: the algebraic formulation of quantum mechanics as well as quantum field theory, and the geometric theory of classical mechanics. These are combined in a unified treatment of the theory of Poisson algebras of observables and pure state spaces with a transition probability, which leads on to a discussion of the theory of quantization and the classical limit from this perspective. A prototype of quantization comes from the analogy between the C*- algebra of a Lie groupoid and the Poisson algebra of the corresponding Lie algebroid. The parallel between reduction of symplectic manifolds in classical mechanics and induced representations of groups and C*- algebras in quantum mechanics plays an equally important role. Examples from physics include constrained quantization, curved spaces, magnetic monopoles, gauge theories, massless particles, and $theta$- vacua. Accessible to mathematicians with some prior knowledge of classical and quantum mechanics, and to mathematical physicists and theoretical physicists with some background in functional analysis.

Quantum Computing and Information
Quantum Computing and Information

Author: Dr. Peter Y. Lee

Publisher: Polaris QCI Publishing

Published: 2024-03-14

Total Pages: 508

ISBN-13: 1961880032

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Unlock the Potential of Quantum Computing This expertly crafted guide demystifies the complexities of quantum computing through a progressive teaching method, making it accessible to students and newcomers alike. Features Explores quantum systems, gates and circuits, entanglement, algorithms, and more. Unique 'scaffolding approach' for easy understanding. Ideal for educators, students, and self-learners. Authors Dr. Peter Y. Lee (Ph.D., Princeton University) – Expert in quantum nanostructures, extensive teaching experience. Dr. Huiwen Ji (Ph.D., Princeton University) – Solid background in quantum chemistry, award-winning researcher. Dr. Ran Cheng (Ph.D., University of Texas at Austin) – Specializes in condensed matter theory, award-winning physicist.

Topological Quantum Computation
Topological Quantum Computation

Author: Zhenghan Wang

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 134

ISBN-13: 0821849301

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Topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. In this approach, information is stored in the lowest energy states of many-anyon systems and processed by braiding non-abelian anyons. The computational answer is accessed by bringing anyons together and observing the result. Besides its theoretical esthetic appeal, the practical merit of the topological approach lies in its error-minimizing hypothetical hardware: topological phases of matter are fault-avoiding or deaf to most local noises, and unitary gates are implemented with exponential accuracy. Experimental realizations are pursued in systems such as fractional quantum Hall liquids and topological insulators. This book expands on the author's CBMS lectures on knots and topological quantum computing and is intended as a primer for mathematically inclined graduate students. With an emphasis on introducing basic notions and current research, this book gives the first coherent account of the field, covering a wide range of topics: Temperley-Lieb-Jones theory, the quantum circuit model, ribbon fusion category theory, topological quantum field theory, anyon theory, additive approximation of the Jones polynomial, anyonic quantum computing models, and mathematical models of topological phases of matter.