Entanglement in Spin Chains: From Theory to Quantum Technology Applications

Preface
References
Contents
Editors and Contributors
About the Editors
Contributors
Entanglement Spectra of Spin Chains
1 Entanglement Spectra of Many-Body Ground States
2 Decomposition of Spin Chain Hilbert Spaces
3 Gapped Spin Chains
4 Gapless Spin Chains
5 Conclusion
References
Detecting Quantum Phase Transitions in Spin Chains
1 Introduction to Quantum Entanglement
1.1 Quantum Entanglement and Quantum Phase Transitions
1.2 Methodologies from the Viewpoint of Quantum Information Theory
1.3 Open Questions
2 Concurrence and Quantum Phase Transitions in Spin Chains
3 von Neumann Entropy and Quantum Phase Transitions
3.1 Single-Site Entanglement
3.2 Multisite Entanglement
3.3 Entanglement and Quantum Phase Transitions at Finite Temperatures
3.4 Entanglement and Quantum Correlations
4 Quantum Discord, Quantum Coherence, and Quantum Phase Transitions
4.1 Quantum Discord
4.2 Quantum Coherence and Quantum Coherence Spectrum
5 Deducing Order Parameters from Entanglement Based Method
6 Summary and Outlook
References
Entanglement Entropy in Critical Quantum Spin Chains with Boundaries and Defects
1 Introduction
2 Entanglement Entropy in CFTs with Boundaries
2.1 Ising Model
2.2 The Free, Compactified Boson Model
3 Entanglement Entropy in CFTs with Defects
3.1 The Ising Model
3.1.1 Energy Defect
3.1.2 Duality Defect
3.2 The Free, Compactified Boson Model
4 Conclusion
References
Entanglement Entropy and Localization in Disordered QuantumChains
1 Introduction
1.1 Generalities
1.2 Random Spin Chain Models
1.2.1 Disordered XXZ Hamiltonians
1.2.2 Random Transverse Field Ising Chains
1.2.3 Many-Body Localization
1.3 Chapter Organization
2 Entanglement in Non-interacting Anderson Localized Chains
2.1 Disordered XX Chains and Single-Particle Localization Lengths
2.1.1 Localization Length from the Participation Ratio (PR)
2.1.2 Numerical Results for the Localization Lengths
2.2 Entanglement Entropy for Many-Body (Anderson Localized) Eigenstates
2.2.1 Free-Fermion Entanglement Entropy
2.2.2 Low and High Energy
2.2.3 Strong Disorder Limit
3 Entanglement and Infinite Randomness Criticalities
3.1 Entanglement in Disordered XXZ and Quantum Ising Chains
3.1.1 Random Singlet State for Disordered S=1/2 Chains
3.1.2 Infinite Randomness Criticality at High Energy
3.2 Other Systems Showing Infinite Randomness Criticality
3.2.1 Higher Spins, Golden Chain, and RG Flows
3.2.2 d>1 Infinite Randomness
3.3 Engineered Disorders
4 Many-Body Localization Probed by Quantum Entanglement
4.1 Area vs. Volume-Law Entanglement for High-Energy Eigenstates
4.2 Distributions of Entanglement Entropies
4.2.1 Distribution Across the ETH-MBL Transition
4.2.2 Strong Disorder Distributions
5 Concluding Remarks
References
Some Aspects of Affleck–Kennedy–Lieb–Tasaki Models: Tensor Network, Physical Properties, Spectral Gap, Deformation, and Quantum Computation
1 Introduction
2 Tensor-Network Picture: MPS and PEPS
2.1 1D AKLT Chain
2.2 Two Dimensions
2.2.1 Honeycomb/Hexagonal Lattice
2.2.2 Square Lattice
2.3 Boundary Conditions and Degeneracy of AKLT Models
3 Magnetic Ordering
4 Symmetry-Protected Topological Order
4.1 SPT Order of 1D AKLT State
4.2 Two Dimensions: Honeycomb and Square Lattices
5 Hidden Order in AKLT States
5.1 String Order Parameter
5.2 Hidden Cluster Order
5.3 Hidden Frustration on Frustrated Lattices
6 Applications in Quantum Computation
6.1 One Dimension
6.1.1 Logical Identity and One-Qubit Gates
6.1.2 Reduction to the 1D Cluster State
6.2 Two Dimensions: Universal Computation
7 Spectral Gap for AKLT Models
8 Deformed AKLT Models and Phase Transitions
8.1 1D Deformed AKLT Chain
8.2 2D Deformed AKLT Models and Their Phase Transitions
9 Conclusion
References
Machine Learning-Assisted Entanglement Measurement in Quantum Many-Body Systems
1 Introduction
2 PPT Criterion and Entanglement Measurement
2.1 Werner States
3 Measuring the PT Moments
3.1 Measurement in Spin Systems
3.2 Measurement in Bosonic Systems
4 Neural Network Entanglement Estimator
4.1 Choice of the Training Set
4.2 Sensitivity and Error Analysis
4.3 Comparison with Approximate State Reconstruction Methods
5 Numerical Results
5.1 Ground States Through a Quantum Phase Transition
5.2 Quench Across a Phase Transition
5.3 W-State
6 Conclusions
References
Local Convertibility in Quantum Spin Systems
1 The Cluster-Ising Model
2 The λ-D Model
3 The Perturbed Toric Code
4 The Quantum Ising Chain
4.1 The Rényi Entropies
4.2 The Correlation Matrix
4.3 The Z2 Symmetric Ground State
4.4 Symmetry Broken Ground State
5 Origin of SSB
5.1 Two-Body Quantum Correlations
5.2 Global Properties: Local Convertibility and Many-Body Entanglement Sharing
5.3 Many-Body Entanglement Distribution
6 Conclusions
Bibliography
Optimal Parent Hamiltonians for Many-Body States
1 Introduction
2 The Space of Symmetries
3 From Ground States to Parent Hamiltonians
4 The Time-Dependent Inverse Problem
5 Conclusions
References
Entanglement Dynamics in Hybrid Quantum Circuits
1 Introduction
2 Random Unitary Quantum Circuits
2.1 Entanglement Growth
2.1.1 Mapping to KPZ Dynamics of Random Surface Growth
2.1.2 Directed Polymer and Minimal-Cut Interpretation
2.2 Operator Spreading
2.3 U(1) Symmetric Circuits
3 Measurement-Induced Phase Transitions
3.1 Entanglement Transition
3.2 Alternative Perspectives on MIPTs
3.2.1 Purification Transition
3.2.2 Ancilla Probe of Purification Transition
3.2.3 Experimental Observation of MIPT in Trapped Ions
3.2.4 Connection to Quantum Channel Capacity and Quantum Error Correction
3.2.5 Information Gained by the Observer
4 Replica Statistical Mechanics Models
4.1 Replica Trick
4.2 Haar Calculus and Boltzmann Weights
4.3 Boundary Conditions and Domain Wall Free Energy
4.4 Symmetry and Conformal Invariance
4.5 Large Hilbert Space Dimension Limit
4.5.1 Mapping Onto Classical Percolation
4.5.2 Entanglement and Minimal-Cut Picture
4.6 Finite d Universality Class
5 Symmetry and Topology in Measurement-Induced Phases and Criticality
5.1 Symmetric Monitored Random Circuits
5.2 Area-Law Phases
5.2.1 Measurement-Induced Symmetry-Breaking Order in 1+1d
5.2.2 Measurement-Induced Topological Orders
5.3 Volume-Law Phases
5.3.1 Volume-Law Phases with Order—Stat-Mech Perspective
5.3.2 Charge Sharpening Transitions in the Volume-Law Phase
6 Discussion
References
Quantum Simulation Using Noisy Unitary Circuits and Measurements
1 Introduction
2 Measurement-Induced Entanglement Transitions in Hybrid Quantum Circuits
2.1 Quantum Trajectories
2.2 Monitored Quantum Circuits
2.3 Purification Transition
2.4 Transitions in the Rényi Entropies
2.5 Analytically Tractable Limits
2.6 Critical Properties of Measurement-Induced Transitions
2.7 Entanglement Transitions in Experiments
2.7.1 Scalability Issues
2.7.2 Measurement-Induced Transition in a Trapped-Ion Experiment
3 Random Circuits on Noisy-Intermediate Scale Quantum Devices
3.1 Random-Circuit Sampling for Achieving a Quantum Computational Advantage
3.2 Applications of Random Circuits in Quantum Many-Body Physics
4 Conclusion
References
Entanglement Dynamics in Spin Chains with Structured Long-Range Interactions
1 Introduction
2 Quantifying Entanglement and Information Spreading
2.1 Measures of Entanglement Entropy
2.2 Lieb–Robinson Bounds and OTOCs
2.3 Quasiparticle Approaches
2.4 Matrix Product States (MPS)
3 Power-Law Interacting Models
3.1 Short-Range Regime, α> 2
3.2 Intermediate Range Regime, 1 < α< 2
3.3 Long-Range Regime, α< 1
4 Fast Scrambling and Sparse Models
4.1 Sparse Nonlocal Interactions for Fast Scrambling
4.2 Sparse Nonlocal Fast Scramblers
5 Implementation in Experiments
5.1 Long-Range Interactions with Trapped Ions
5.2 Long-Range Interactions with Rydberg Atoms
5.3 Long-Range Interactions in Cavity Quantum Electrodynamics
6 Outlook and Further Connections
References
Quantum Map Approach to Entanglement Transfer and Generation in Spin Chains
1 Introduction
2 Quantum Dynamical Maps
3 U(1)-Symmetric Hamiltonians
4 One-Qubit Map
5 Two-Qubit Map
6 Two-Qubit Entanglement Generation
7 Four-Qubit Entanglement Generation
8 Conclusion
References
Weak Ergodicity Breaking Through the Lens of QuantumEntanglement
1 Introduction
2 Matrix Product State Methods
2.1 Towers of Quasiparticles
2.2 Time-Dependent Variational Principle
3 Mechanisms of Weak Ergodicity Breaking
3.1 Spectrum Generating Algebra
3.2 Hilbert Space Fragmentation
3.3 Projector Embedding
4 PXP Model
4.1 The Model
4.2 Ergodicity Breaking in the PXP Model
4.3 The Origin of Non-thermal Eigenstates and Quantum Revivals
5 Semiclassical Dynamics
5.1 Discussion: Benefits and Pitfalls of TDVP
6 Quantum Many-Body Scars
6.1 Scars in Few-Body Systems
6.2 Quasimodes in the PXP Model
6.3 Discussion: Scars or Not?
7 Weak Ergodicity Breaking in Experiment
7.1 Rydberg Atoms
7.2 Tilted Optical Lattices
8 Conclusions
References
Quench Dynamics of Rényi Negativities and the Quasiparticle Picture
1 Introduction
2 Entanglement Measures for Mixed States
2.1 Entanglement in Mixed States and Logarithmic Negativity
2.2 Entanglement Detection Through Partial Transpose Moments
3 Quench Dynamics of Rényi Negativities in Conformal Field Theory
3.1 Rényi Negativities from Twist Field Correlation Functions
3.2 Out-of-Equilibrium Dynamics of the Rényi Negativities
4 Quasiparticle Picture for the Rényi Negativities in Integrable Systems
4.1 Quasiparticle Picture
4.2 The Quasiparticle Description for Rényi Negativities
5 Time Evolution of Rényi Negativities in Free Models: Numerical Results
5.1 Mass Quench in the Harmonic Chain
5.2 Quench in a Free-Fermion Chain
5.3 Quasiparticle Prediction for the pn-PPT Conditions
6 Conclusions
References
Phases and Dynamics of Ultracold Bosons in a Tilted Optical Lattice
1 Introduction
2 Experimental Platforms
3 Model and Phases
4 Non-equilibrium Dynamics
4.1 Quench and Ramp Protocols
4.2 Periodic Protocols
4.2.1 Methods
4.2.2 Results
5 Hilbert Space Fragmentation: A Minimal Model
5.1 Model and Fragmentation
5.2 Adding a Staggered Field
5.3 Connection to Lattice Gauge Theories
6 Discussion
References
NMR Experimental Study of Out-of-Equilibrium Spin Models
1 Nuclear Magnetic Resonance
1.1 Nuclear Magnetic Resonance in Condensed Matter
1.1.1 NMR Hamiltonian
1.1.2 Equilibrium States
1.1.3 Dynamics
1.1.4 Control by Radio-Frequency Fields
1.2 Nuclear Spin Systems
1.2.1 3D Spin Systems
1.2.2 Quasi-1D Spin Systems
1.2.3 Finite-Size Systems
2 Control Techniques for Hamiltonian Engineering
2.1 Average Hamiltonian Design
2.2 Magnus Expansion and Average Hamiltonian Theory
2.3 Floquet-Magnus Expansion
2.3.1 Convergence of the Expansions
2.3.2 Examples of Sequences
3 State and Observable Preparation
3.1 Phase Cycling
3.2 Dipolar Order State
3.2.1 Jeener-Broekaert Pulse Pair
3.2.2 Adiabatic Demagnetization in the Rotating Frame
3.2.3 Verification and Extension
3.3 Local Observables
4 Time-Ordered Correlations
4.1 Echoes and Fidelity Decay
4.2 Two-Point Correlators
5 Out-of-Time-Ordered Correlations
5.1 Multiple Quantum Coherences and Spin Counting
5.2 OTO Commutator and Multiple Quantum Coherences
5.3 Entanglement Measures
5.3.1 OTOC with Full Control to Detect the Rényi Entropy
5.3.2 Correlation Rényi Entropies
5.4 Approximate Rényi Entropy and Average Correlation Length
6 Integrable Models
6.1 Kicked Evolution
6.2 Quantum State Transfer
7 Non-Integrable Models
7.1 Many-Body Localization
7.2 Prethermal Regime
References
Quantum-Dot Spin Chains
1 Introduction
2 Gate-Defined Quantum Dots
2.1 Quantum-Dot Fabrication
2.2 Quantum-Dot Operation
2.3 Exchange Coupling in Quantum Dots
3 Quantum Information Processing with Exchange-Coupled Quantum-Dot Spins
3.1 Single-Spin Qubits
3.2 Two-Spin Qubits
3.3 Three-Spin Qubits
4 Quantum Simulation with Quantum-Dot Spin Chains
4.1 Charge Physics in the Hubbard Model
4.2 Spin Physics in the Hubbard Model
4.3 Spin Physics in the Heisenberg Model
5 Quantum State Transfer in Spin Chains
6 Future Directions and Outlook
References
Index