Quantum information meets quantum matter : from quantum entanglement to topological phases of many-body systems
Bei Zeng (Author), Xie Chen (Author), Duan-Lu Zhou (Author), Xiao-Gang Wen (Author)
This book approaches condensed matter physics from the perspective of quantum information science, focusing on systems with strong interaction and unconventional order for which the usual condensed matter methods like the Landau paradigm or the free fermion framework break down. Concepts and tools in quantum information science such as entanglement, quantum circuits, and the tensor network representation prove to be highly useful in studying such systems. The goal of this book is to introduce these techniques and show how they lead to a new systematic way of characterizing and classifying quantum phases in condensed matter systems. The first part of the book introduces some basic concepts in quantum information theory which are then used to study the central topic explained in Part II: local Hamiltonians and their ground states. Part III focuses on one of the major new phenomena in strongly interacting systems, the topological order, and shows how it can essentially be defined and characterized in terms of entanglement. Part IV shows that the key entanglement structure of topological states can be captured using the tensor network representation, which provides a powerful tool in the classification of quantum phases. Finally, Part V discusses the exciting prospect at the intersection of quantum information and condensed matter physics - the unification of information and matter. Intended for graduate students and researchers in condensed matter physics, quantum information science and related fields, the book is self-contained and no prior knowledge of these topics is assumed
Springer eBooks
Electronic books
1 online resource (xxii, 364 pages) : illustrations (some color)
9781493990849, 9781493990825, 9781493990832, 1493990845, 1493990829, 1493990837
1091358969
Printed edition:
Intro; Foreword; Preface; Structure of the Book; Acknowledgements; Contents; Part I Basic Concepts in Quantum Information Theory; 1 Correlation and Entanglement; 1.1 Introduction; 1.2 Correlations in Classical Probability Theory; 1.2.1 Joint Probability Without Correlations; 1.2.2 Correlation Functions; 1.2.3 Mutual Information; 1.3 Quantum Entanglement; 1.3.1 Pure and Mixed Quantum States; 1.3.2 Composite Quantum Systems and Tensor Product Structure; 1.3.3 Pure Bipartite State, Schmidt Decomposition; 1.3.4 Mixed Bipartite State; 1.3.5 Bell's Inequalities; 1.3.6 Entanglement 1.4 Correlation and Entanglement in Many-Body Quantum Systems1.4.1 The GHZ Paradox; 1.4.2 Many-Body Correlation; 1.4.3 Many-Body Entanglement; 1.5 Summary and Further Reading; References; 2 Evolution of Quantum Systems; 2.1 Introduction; 2.2 Unitary Evolution; 2.2.1 Single Qubit Unitary; 2.2.2 Two-Qubit Unitary; 2.2.3 N-Qubit Unitary; 2.3 Quantum Circuits; 2.4 Open Quantum Systems; 2.5 Master Equation; 2.5.1 The Lindblad Form; 2.5.2 Master Equations for a Single Qubit; 2.6 Summary and Further Reading; References; 3 Quantum Error-Correcting Codes; 3.1 Introduction 3.2 Basic Idea of Error Correction3.2.1 Bit Flip Code; 3.2.2 Shor's Code; 3.2.3 Other Noise Models; 3.3 Quantum Error-Correcting Criteria, Code Distance; 3.4 The Stabilizer Formalism; 3.4.1 Shor's Code; 3.4.2 The Stabilizer Formalism; 3.4.3 Stabilizer States and Graph States; 3.5 Toric Code; 3.6 Summary and Further Reading; References; Part II Local Hamiltonians, Ground States, and Many-Body Entanglement; 4 Local Hamiltonians and Ground States; 4.1 Introduction; 4.2 Many-Body Hilbert Space; 4.3 Local Hamiltonians; 4.3.1 Examples; 4.3.2 The Effect of Locality 4.4 Ground-State Energy of Local Hamiltonians4.4.1 The Local Hamiltonian Problem; 4.4.2 The Quantum Marginal Problem; 4.4.3 The N-Representability Problem; 4.4.4 de Finetti Theorem and Mean-Field Bosonic Systems; 4.5 Frustration-Free Hamiltonians; 4.5.1 Examples of Frustration-Free Hamiltonians; 4.5.2 The Frustration-Free Hamiltonians Problem; 4.5.3 The 2-Local Frustration-Free Hamiltonians; 4.6 Summary and Further Reading; References; 5 Gapped Quantum Systems and Entanglement Area Law; 5.1 Introduction; 5.2 Quantum Many-Body Systems; 5.2.1 Dimensionality and Locality