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E-raamat: Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems

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Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body SystemsSuurem pilt
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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.

Arvustused

Quantum information meets quantum matter is bound to hold an honored place on the bookshelves of many scientists for years to come. From myself, I would add that of students and PhD students, I do believe! (Eugene Kryachko, zbMATH 1423.81010, 2019)

Part I Basic Concepts in Quantum Information Theory
1 Correlation and Entanglement
3(34)
1.1 Introduction
3(1)
1.2 Correlations in Classical Probability Theory
4(9)
1.2.1 Joint Probability Without Correlations
5(3)
1.2.2 Correlation Functions
8(2)
1.2.3 Mutual Information
10(3)
1.3 Quantum Entanglement
13(13)
1.3.1 Pure and Mixed Quantum States
13(4)
1.3.2 Composite Quantum Systems and Tensor Product Structure
17(2)
1.3.3 Pure Bipartite State, Schmidt Decomposition
19(2)
1.3.4 Mixed Bipartite State
21(1)
1.3.5 Bell's Inequalities
22(1)
1.3.6 Entanglement
23(3)
1.4 Correlation and Entanglement in Many-Body Quantum Systems
26(7)
1.4.1 The GHZ Paradox
27(1)
1.4.2 Many-Body Correlation
28(3)
1.4.3 Many-Body Entanglement
31(2)
1.5 Summary and Further Reading
33(1)
References
34(3)
2 Evolution of Quantum Systems
37(26)
2.1 Introduction
37(2)
2.2 Unitary Evolution
39(7)
2.2.1 Single Qubit Unitary
39(2)
2.2.2 Two-Qubit Unitary
41(2)
2.2.3 N-Qubit Unitary
43(3)
2.3 Quantum Circuits
46(4)
2.4 Open Quantum Systems
50(2)
2.5 Master Equation
52(7)
2.5.1 The Lindblad Form
52(2)
2.5.2 Master Equations for a Single Qubit
54(5)
2.6 Summary and Further Reading
59(2)
References
61(2)
3 Quantum Error-Correcting Codes
63(22)
3.1 Introduction
63(1)
3.2 Basic Idea of Error Correction
64(5)
3.2.1 Bit Flip Code
64(3)
3.2.2 Shor's Code
67(1)
3.2.3 Other Noise Models
68(1)
3.3 Quantum Error-Correcting Criteria, Code Distance
69(3)
3.4 The Stabilizer Formalism
72(6)
3.4.1 Shor's Code
73(3)
3.4.2 The Stabilizer Formalism
76(1)
3.4.3 Stabilizer States and Graph States
77(1)
3.5 Toric Code
78(2)
3.6 Summary and Further Reading
80(2)
References
82(3)
Part II Local Hamiltonians, Ground States, and Many-Body Entanglement
4 Local Hamiltonians and Ground States
85(30)
4.1 Introduction
85(3)
4.2 Many-Body Hilbert Space
88(1)
4.3 Local Hamiltonians
89(3)
4.3.1 Examples
90(1)
4.3.2 The Effect of Locality
91(1)
4.4 Ground-State Energy of Local Hamiltonians
92(12)
4.4.1 The Local Hamiltonian Problem
93(2)
4.4.2 The Quantum Marginal Problem
95(4)
4.4.3 The N-Representability Problem
99(1)
4.4.4 De Finetti Theorem and Mean-Field Bosonic Systems
100(4)
4.5 Frustration-Free Hamiltonians
104(6)
4.5.1 Examples of Frustration-Free Hamiltonians
104(2)
4.5.2 The Frustration-Free Hamiltonians Problem
106(1)
4.5.3 The 2-Local Frustration-Free Hamiltonians
107(3)
4.6 Summary and Further Reading
110(2)
References
112(3)
5 Gapped Quantum Systems and Entanglement Area Law
115(42)
5.1 Introduction
115(2)
5.2 Quantum Many-Body Systems
117(6)
5.2.1 Dimensionality and Locality
117(1)
5.2.2 Thermodynamic Limit and Universality
118(1)
5.2.3 Gap
118(2)
5.2.4 Correlation
120(2)
5.2.5 Entanglement
122(1)
5.3 Entanglement Area Law in Gapped Systems
123(6)
5.3.1 Entanglement Area Law
123(2)
5.3.2 Topological Entanglement Entropy
125(4)
5.4 Generalizations of Topological Entanglement Entropy
129(17)
5.4.1 Quantum Conditional Mutual Information
129(3)
5.4.2 Toric Code in a Magnetic Field
132(4)
5.4.3 The Transverse-Field Ising Model
136(3)
5.4.4 The Transverse-Field Cluster Model
139(4)
5.4.5 Systems with Mixed Orders
143(1)
5.4.6 I(A:C\B) as a Detector of Nontrivial Many-Body Entanglement
143(3)
5.5 Gapped Ground States as Quantum Error-Correcting Codes
146(2)
5.6 Entanglement in Gapless Systems
148(2)
5.7 Summary and Further Reading
150(2)
References
152(5)
Part III Topological Order and Long-Range Entanglement
6 Introduction to Topological Order
157(34)
6.1 Introduction
157(5)
6.1.1 Phases of Matter and Landau's Symmetry-Breaking Theory
157(2)
6.1.2 Quantum Phases of Matter and Transverse-Field Ising Model
159(1)
6.1.3 Physical Ways to Understand Symmetry Breaking in Quantum Theory
160(2)
6.1.4 Compare a Finite-Temperature Phase with a Zero-Temperature Phase
162(1)
6.2 Topological Order
162(2)
6.2.1 The Discovery of Topological Order
162(2)
6.3 A Macroscopic Definition of Topological Order
164(3)
6.3.1 What is "Topological Ground-State Degeneracy"
166(1)
6.3.2 What is "Non-Abelian Geometric Phase of Topologically Degenerate States"
167(1)
6.4 A Microscopic Picture of Topological Orders
167(4)
6.4.1 The Essence of Fractional Quantum Hall States
167(1)
6.4.2 Intuitive Pictures of Topological Order
168(3)
6.5 What is the Significance of Topological Order?
171(1)
6.6 Quantum Liquids of Unoriented Strings
172(2)
6.7 The Emergence of Fractional Quantum Numbers and Fermi/Fractional Statistics
174(2)
6.7.1 Emergence of Fractional Angular Momenta
174(1)
6.7.2 Emergence of Fermi and Fractional Statistics
175(1)
6.8 Topological Degeneracy of Unoriented String Liquid
176(2)
6.9 Topological Excitations and String Operators
178(7)
6.9.1 Toric Code Model and String Condensation
178(2)
6.9.2 Local and Topological Excitations
180(1)
6.9.3 Three Types of Quasiparticles
181(1)
6.9.4 Three Types of String Operators
181(3)
6.9.5 Statistics of Ends of Strings
184(1)
6.10 Summary and Further Reading
185(1)
References
186(5)
7 Local Transformations and Long-Range Entanglement
191(42)
7.1 Introduction
191(1)
7.2 Quantum Phases and Phase Transitions
192(3)
7.3 Quantum Phases and Local Unitary Transformations
195(8)
7.3.1 Quantum Phases and Local Unitary Evolutions in Ground States
195(2)
7.3.2 Local Unitary Evolutions and Local Unitary Quantum Circuits
197(3)
7.3.3 Local Unitary Quantum Circuits and Wave Function Renormalization
200(3)
7.4 Gapped Hamiltonians and Topological Order
203(7)
7.4.1 Gapped Quantum Systems and Gapped Quantum Phases
204(1)
7.4.2 Gapped Quantum Liquid System and Gapped Quantum Liquid Phase
205(3)
7.4.3 Topological Order
208(2)
7.5 Universality Classes of Many-Body Wave Functions
210(11)
7.5.1 Gapped Quantum Liquid
210(2)
7.5.2 Symmetry-Breaking Order
212(2)
7.5.3 Stochastic Local Transformations and Long-Range Entanglement
214(7)
7.6 Symmetry-Protected Topological Order
221(2)
7.7 A New
Chapter in Physics
223(1)
7.8 Summary and Further Reading
224(2)
References
226(7)
Part IV Gapped Topological Phases and Tensor Networks
8 Matrix Product State and ID Gapped Phases
233(22)
8.1 Introduction
233(1)
8.2 Matrix Product States
234(13)
8.2.1 Definition and Examples
234(2)
8.2.2 Double Tensor
236(2)
8.2.3 Calculation of Norm and Physical Observables
238(1)
8.2.4 Correlation Length
239(1)
8.2.5 Entanglement Area Law
240(1)
8.2.6 Gauge Degree of Freedom
241(1)
8.2.7 Projected Entangled Pair Picture
242(1)
8.2.8 Canonical Form
243(1)
8.2.9 Injectivity
244(2)
8.2.10 Parent Hamiltonian
246(1)
8.3 Renormalization Group Transformation on MPS
247(2)
8.4 No Intrinsic Topological Order in ID Bosonic Systems
249(3)
8.5 Summary and Further Reading
252(1)
References
253(2)
9 Tensor Product States and 2D Gapped Phases
255(26)
9.1 Introduction
255(1)
9.2 Tensor Product States
256(7)
9.2.1 Definition and Examples
256(3)
9.2.2 Properties
259(4)
9.3 Tensor Network for Symmetry-Breaking Phases
263(3)
9.3.1 Ising Model
264(1)
9.3.2 Structural Properties
264(1)
9.3.3 Symmetry Breaking and the Block Structure of Tensors
265(1)
9.4 Tensor Network for Topological Phases
266(9)
9.4.1 Toric Code Model
267(1)
9.4.2 Structural Properties
267(2)
9.4.3 Topological Property from Local Tensors
269(2)
9.4.4 Stability Under Symmetry Constraint
271(4)
9.5 Other Forms of Tensor Network Representation
275(2)
9.5.1 Multiscale Entanglement Renormalization Ansatz
275(1)
9.5.2 Tree Tensor Network State
276(1)
9.6 Summary and Further Reading
277(1)
References
278(3)
10 Symmetry-Protected Topological Phases
281(54)
10.1 Introduction
281(1)
10.2 Symmetry-Protected Topological Order in ID Bosonic Systems
282(21)
10.2.1 Examples
282(3)
10.2.2 On-Site Unitary Symmetry
285(9)
10.2.3 Time Reversal Symmetry
294(2)
10.2.4 Translation Invariance
296(7)
10.2.5 Summary of Results for Bosonic Systems
303(1)
10.3 Topological Phases in ID Fermion Systems
303(7)
10.3.1 Jordan-Wigner Transformation
304(1)
10.3.2 Fermion Parity Symmetry Only
305(2)
10.3.3 Fermion Parity and T2 = 1 Time Reversal
307(1)
10.3.4 Fermion Parity and T2 ≠ 1 Time Reversal
308(1)
10.3.5 Fermion Number Conservation
309(1)
10.4 2D Symmetry-Protected Topological Order
310(15)
10.4.1 2D AKLT Model
310(4)
10.4.2 2D CZX Model
314(11)
10.5 General Construction of SPT Phases
325(4)
10.5.1 Group Cohomology
325(2)
10.5.2 SPT Model from Group Cohomology
327(2)
10.6 Summary and Further Reading
329(1)
References
330(5)
Part V Outlook
11 A Unification of Information and Matter
335
11.1 Four Revolutions in Physics
335(9)
11.1.1 Mechanical Revolution
336(2)
11.1.2 Electromagnetic Revolution
338(1)
11.1.3 Relativity Revolution
339(3)
11.1.4 Quantum Revolution
342(2)
11.2 It from Qubit, Not Bit
344(3)
11.3 Emergence Approach
347(14)
11.3.1 Two Approaches
347(2)
11.3.2 Principle of Emergence
349(3)
11.3.3 String-Net Liquid of Qubits Unifies Light and Electrons
352(5)
11.3.4 Evolving Views for Light and Gauge Theories
357(2)
11.3.5 Where to Find Long-Range Entangled Quantum Matter?
359(2)
References
361
Bei Zeng received the B.Sc. degree in physics and mathematics and M.Sc. degree in physics from Tsinghua University, Beijing, China, in 2002 and 2004, respectively. She received the Ph.D. degree in physics from Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts, USA, in 2009. From 2009 to 2010, she was a postdoctoral fellow at the Institute for Quantum Computing (IQC) and the Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, Canada. In 2010, she joined the Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario, Canada, as an assistant professor, and promoted to Tenured Associate Professor in 2014 and Professor in 2018.





 





Xie Chen is an Associate Professor of Theoretical Physics at the California Institute of Technology. Dr. Chen obtained her Ph.D. degree from MIT in 2012 and was a Miller research fellow at the University of California, Berkeley for two yearsbefore joining Caltech in 2014. Dr. Chen is interested in studying quantum mechanical systems with a large number of degrees of freedom and how the constituent degrees of freedom cooperate with each other to realize amazing emergent phenomena, both at equilibrium and in dynamical processes. Dr. Chen was awarded the Sloan Research Fellowship and the NSF Faculty Early Career Award in 2017.





 





Duan-Lu Zhou is a Professor of physics at the Institute of Physics, Beijing National Laboratory for Condensed Matter, Physics, Chinese Academy of Sciences, and also teaches at University of Chinese Academy of Sciences. His main research interest is in quantum physics and quantum information, where he has published 70 academic papers.





 





Xiao-Gang Wen is a theoretical condensed matter physicist, recognized for his work on introducing the notion topological order (1989) and developing the theories of this new class of quantum states of matter. He is known particularly for his theoretical discovery of perfect conducting 1D chiral Luttinger liquids, Z2 spin liquid with emergent fermion, non-abelian fractional quantum Hall states for topological quantum computation, symmetry protected topological order, and string-net unification of elementary particles and interactions. Since 2000, the study of topological states of matter slowly became a very active new field in condensed matter physics. Wen enter the graduate school of Princeton University in 1982, and earned a Ph.D degree in the field of superstring theory under Prof. Witten. During his postdoctoral period (1987-1989) in ITP, Santa Barbara, he started to pursue research in condensed matter physics. After a two-years stay in IAS, Princeton, he joined the faculty of department of Physics, MIT in 1991. He was a Distinguished Moore Scholar at Caltech (2006). Newton Chair at Perimeter Institute for Theoretical Physics (2012 2014). He was awarded Oliver E. Buckley Condensed Matter Prize by APS in 2017, and Dirac Medal by ICTP in 2018. He is a Cecil and Ida Green Professor of Physics at MIT since 2004, and became a member of National Academy of Science in 2018.
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