Frontiers in Quantum Information Research : Decoherence, Entanglement, Entropy, MPS and DMRG.

By: Nakahara, MikioContributor(s): Tanaka, ShuMaterial type: TextTextSeries: Kinki University Series on Quantum Computing SerPublisher: Singapore : World Scientific Publishing Co Pte Ltd, 2012Copyright date: ©2012Description: 1 online resource (359 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789814407199Subject(s): Information display systems -- Congresses | Quantum electronics -- CongressesGenre/Form: Electronic books.Additional physical formats: Print version:: Frontiers in Quantum Information Research : Decoherence, Entanglement, Entropy, MPS and DMRGDDC classification: 004.1 LOC classification: QC688 -- .F76 2012ebOnline resources: Click to View
Contents:
Intro -- CONTENTS -- Preface -- Summer School on Decoherence, Entanglement and Entropy Oxford Kobe Institute (Kobe, Japan) -- Workshop on Matrix Product State Formulation and Density Matrix Renormalization Group Simulations (MPS&DMRG) Oxford Kobe Institute (Kobe, Japan) -- List of Participants -- Committees -- Part A Summer School on Decoherence, Entanglement and Entropy -- Black Holes and Qubits L. Borsten, M. J. Du., and W. Rubens -- Overview -- 1. Qubits and entanglement -- 1.1. A brief introduction to quantum information -- 1.1.1. Qubits -- 1.2. Entanglement and the Bell inequality -- 1.2.1. Entanglement dependent quantum information -- 1.3. Entanglement classification -- 1.3.1. Bell inequalities without the inequality -- 1.3.2. The SLOCC paradigm -- 1.3.3. Entanglement measures -- 1.3.4. Stochastic LOCC equivalence -- 1.4. Bipartite entanglement -- 1.4.1. Generic finite-dimensional bipartite systems -- 1.4.2. Two qubits -- 1.5. Three qubit entanglement -- 1.5.1. Local unitary invariants -- 1.5.2. Cayley's hyperdeterminant -- 1.5.3. Entanglement classification -- 2. Black holes in M-theory -- 2.1. The road to M-theory -- 2.2. Black holes -- 2.2.1. Extremal black holes -- 2.3. Black hole thermodynamics -- 2.4. Black holes in supergravity -- 2.5. The STU model -- 2.5.1. The Lagrangian -- 2.5.2. The Bogomol'nyi spectrum -- 2.5.3. Black hole entropy -- 3. STU black holes and three qubits -- 3.1. Entropy/entanglement correspondence -- 3.2. Rebits -- 3.3. Classification of N = 2 black holes and three-qubit states -- 3.4. Further developments -- 3.4.1. Microscopic interpretation -- 3.4.2. 4-qubit entanglement and the STU model in D = 3 -- 4. Beyond the STU model -- 4.1. N = 8 supergravity and black holes -- 5. E7 and the tripartite entanglement of seven qubits -- 6. Fano plane entanglement and the octonions -- 6.1. Composition algebras.
6.2. The octonionic tripartite entanglement of seven qubits -- 6.3. Subsectors -- 7. Cubic Jordan algebras and the Freudenthal triple system -- 7.1. Cubic Jordan algebras -- 7.2. The Freudenthal triple system -- 8. The 3-qubit Freudenthal triple system -- 8.1. The FTS rank entanglement classes -- 8.1.1. Rank 1 and the class of separable states -- 8.1.2. Rank 2 and the class of biseparable states -- 8.1.3. Rank 3 and the class of W-states -- 8.1.4. Rank 4 and the class of GHZ-states -- 8.2. SLOCC orbits -- 9. Supersymmetric quantum information -- 10. Supergroups -- 10.1. Grassmann numbers -- 10.2. Super linear algebra -- 10.3. Orthosymplectic superalgebras -- 11. Super Hilbert space and uOSp(1 2) -- 11.0.1. Physical states -- 11.1. The superqubit -- 12. Super entanglement -- 12.1. Two superqubits -- 12.2. Three superqubits -- Acknowledgments -- References -- Weak Value with Decoherence A. Hosoya -- 1. Introduction -- 2. Weak Value -- 3. Weak Measurement with Decoherence -- 3.1. Weak Measurement-Review -- 3.2. Weak Measurement and Environment -- 4. Geometric Phase -- 5. Summary -- Bibliography -- Lectures on Matrix Product Representation of States V. Karimipour and M. Asoudeh -- 1. Introduction -- Part I: Matrix Product States in Quantum Spin Chains -- 2. Quantum Spin Chains -- 3. Ground States of Spin Chains as Matrix Product States -- 3.1. Construction of the MPS -- 3.2. Correlation Functions -- 3.3. Symmetries of the Ground State -- 4. The Parent Hamiltonian -- 4.1. Symmetries of the Hamiltonian -- 4.2. Some Examples -- 4.2.1. The GHZ state -- 4.2.2. The W state -- 4.3. Cluster States -- 4.4. A Simple Spin One-half Chain with Phase Transition -- 4.5. The Majumdar-Ghosh Models -- 4.5.1. The AKLT states -- 5. Conclusion of Part I -- Part II: One Dimensional Reaction Diffusion Systems -- 6. The Master Equation and its Hamiltonian Form.
7. The Stationary State and it's Matrix Product Form -- 8. An Example: The Asymmetric Simple Exclusion Process -- 8.1. Correlation Functions -- 9. The Phase Diagram -- 10. Extensions of the ASEP -- 11. Conclusion of Part II -- 12. Acknowledgements -- References -- On a Possible Definition of Entanglement in Antisymmetric States T. Ichikawa, T. Sasaki, and I. Tsutsui -- Acknowledgements -- References -- Entanglement Measures for Intermediate Separability T. Sasaki, T. Ichikawa, and I. Tsutsui -- 1. Introduction -- 2. Entanglement Measures as Yardsticks of Separability in Arbitrary Subsystems -- 3. Illustrating Rm for some Classes of Globally Entangled States -- 4. Physical Interpretations of the Measures Rm -- 5. Conclusion -- Acknowledgments -- References -- Unruh Effect on Quantum Teleportation and Entanglement: Implications on Black Hole Information K. Shiokawa -- 1. Introduction -- 2. General Formulation -- 3. Entanglement Dynamics of Accelerated Oscillators -- 3.1. Bipartite entanglement -- 3.2. Tripartite entanglement dynamics -- 4. Quantum Teleportation Dynamics with an Accelerated Observer -- 5. Implications -- 6. Acknowledgments -- References -- Systematic Construction of Generalized Bell Inequalities S. Tanimura -- 1. Introduction -- 2. Bell-Clauser-Horne-Shimony-Holt Inequality -- 2.1. Setting up -- 2.2. Hidden-variable theory -- 2.3. Quantum mechanics -- 2.4. Role of locality -- 3. Why Is the BCHSH Inequality Violated? -- 3.1. Interference effect -- 3.2. Noncommutativity and a useful trick -- 4. New Test -- 4.1. New quantity -- 4.2. Hidden-variable theory vs. quantum mechanics -- 5. Concluding Remarks -- Acknowledgements -- References -- On 3-Variable Exponential Polynomials and Quantum Algorithms Y. Ohno, Y. Sasaki, and C. Yamazaki -- 1. Introduction -- 2. The Number of Solutions of Equation.
3. Calculation of the Deterministic Time for a Classical Algorithm -- 4. Calculation of the Time Complexity for a Quantum Algorithm -- 5. Concluding Remarks -- Acknowledgments -- Appendix A. Algebra -- Appendix A.1. Group -- Appendix A.2. Residue class of N -- Appendix A.3. Ring -- Appendix A.4. Field -- References -- Part B Workshop on Matrix Product State Formulation and Density Matrix Renormalization Group Simulations (MPS&DMRG) -- Application of Density Matrix Renormalization Group Method to Photoinduced Phenomena in Strongly Correlated Electron Systems H. Matsueda -- 1. Introduction -- 1.1. Motivation from condensed matter physics side -- 1.2. Quantum information perspective -- 2. Concept of DMRG: A Modern Approach -- 2.1. Superblock and singular value decomposition -- 2.2. Entanglement entropy -- 2.3. Sweeping and matrix product state -- 2.4. Finite-system sweep of DMRG -- 3. Extention of DMRG for Excited States -- 3.1. Targetting excited states into density matrix -- 3.2. Optical spectra -- 3.2.1. Linear optical absorption and optically forbidden spectrum -- 3.2.2. Kramers-Kronig relation -- 3.2.3. Two correction vectors -- 3.3. Angle resolved photoemission spectroscopy -- 3.3.1. DMRG calculation of ARPES spectra -- 3.3.2. Pseudo-momenta -- 4. Time-dependent DMRG -- 4.1. Direct approach similar to dynamical DMRG -- 4.2. MPS and time evolution -- 4.3. Wave function transformation -- 5. Optical Physics of 1D Mott Insulators -- 5.1. Introduction -- 5.2. Fundamental concepts in photoexcited 1D Mott insulators -- 5.2.1. Photo-carrier doping into 1D systems -- 5.2.2. Physics of spin-charge separation, holon-doublon picture -- 5.2.3. Coupling between photocarriers and phonons -- 5.3. Relevant models for optically-excited 1D Mott insulators -- 5.3.1. Hubbard models -- 5.3.2. Holstein and Peierls couplings -- 5.3.3. Pseudo-phonon method.
5.3.4. Holon-doublon model -- 5.4. Photoexcited and single-particle excited states of 1D Mott insulators -- 5.4.1. Degenerate photoexcited states with even and odd parity -- 5.4.2. Effect of on-site and nearest-neighbor Coulomb interactions on Holstein-type electron-phonon coupling -- 5.4.3. Stability of spin-charge separation in the presence of Holstein-type electron-phonon coupling -- 5.5. Ultrafast relaxation dynamics -- 5.5.1. Motivations and purposes -- 5.5.2. Time-dependent quantities -- 5.5.3. Numerical results and perspectives -- 6. What is Effective Spatial Dimension of Fermion-Boson Coupled System? -- 7. Photoinduced States of Multi-Orbital Systems -- 7.1. Manganites -- 7.2. Cobaltites -- 8. Summary -- Acknowledgement -- References -- Density-Matrix Renormalization Group Method for Tomonaga-Luttinger Liquid T. Hikihara -- 1. Introduction -- 2. Combined Scheme of Bosonization and DMRG Methods -- 2.1. Bosonization approach -- 2.2. DMRG results -- 3. Applications -- 3.1. XXZ chain under magnetic field -- 3.2. Two-leg ladder under magnetic field -- 3.3. Zigzag ladder under magnetic field -- 4. Summary -- Acknowledgments -- References -- Supersymmetric Valence-Bond Solid Models - Hidden Order and Dynamics K. Totsuka and K. Hasebe -- 1. Introduction -- 2. Valence Bond Solid Models -- 3. Supersymmetric Valence Bond Solid Models -- 3.1. UOSp(1 2) supersymmetry -- 3.2. UOSp(1 2) SVBS states -- 3.3. Supersymmetric parent Hamiltonian -- 4. Matrix Product Representation -- 4.1. Matrix-product formalism -- 4.2. SMPS formalism -- 5. Magnetic- and Superconducting Properties: "Bosons v.s. Fermions -- 6. Hidden String Order -- 7. Lowlying Excitations -- 8. SU(2 1) SVBS State -- 9. Summary -- References -- Matrix Product States in Quantum Integrable Models H. Katsura and I. Maruyama -- 1. Introduction -- 2. Algebraic Bethe Ansatz for the Spin-1 2 XXZ Chain.
3. Matrix Product State Representation for the Bethe States in the Spin-1 2 XXZ Chain.
Summary: This book is a collection of lecture notes/contributions from a summer school on decoherence, entanglement & entropy and a workshop on MPS (matrix product states) & DMRG (density matrix renormalization group). Subjects of the summer school include introduction to MPS, black holes, qubits and octonions, weak measurement, entanglement measures and separability, generalized Bell inequalities, among others. Subjects of the workshop are dedicated to MPS and DMRG. Applications to strongly correlated systems and integrable systems are also mentioned. Contributions to this book are prepared in a self-contained manner so that a reader with a modest background in quantum information and quantum computing may understand the subjects.
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Intro -- CONTENTS -- Preface -- Summer School on Decoherence, Entanglement and Entropy Oxford Kobe Institute (Kobe, Japan) -- Workshop on Matrix Product State Formulation and Density Matrix Renormalization Group Simulations (MPS&DMRG) Oxford Kobe Institute (Kobe, Japan) -- List of Participants -- Committees -- Part A Summer School on Decoherence, Entanglement and Entropy -- Black Holes and Qubits L. Borsten, M. J. Du., and W. Rubens -- Overview -- 1. Qubits and entanglement -- 1.1. A brief introduction to quantum information -- 1.1.1. Qubits -- 1.2. Entanglement and the Bell inequality -- 1.2.1. Entanglement dependent quantum information -- 1.3. Entanglement classification -- 1.3.1. Bell inequalities without the inequality -- 1.3.2. The SLOCC paradigm -- 1.3.3. Entanglement measures -- 1.3.4. Stochastic LOCC equivalence -- 1.4. Bipartite entanglement -- 1.4.1. Generic finite-dimensional bipartite systems -- 1.4.2. Two qubits -- 1.5. Three qubit entanglement -- 1.5.1. Local unitary invariants -- 1.5.2. Cayley's hyperdeterminant -- 1.5.3. Entanglement classification -- 2. Black holes in M-theory -- 2.1. The road to M-theory -- 2.2. Black holes -- 2.2.1. Extremal black holes -- 2.3. Black hole thermodynamics -- 2.4. Black holes in supergravity -- 2.5. The STU model -- 2.5.1. The Lagrangian -- 2.5.2. The Bogomol'nyi spectrum -- 2.5.3. Black hole entropy -- 3. STU black holes and three qubits -- 3.1. Entropy/entanglement correspondence -- 3.2. Rebits -- 3.3. Classification of N = 2 black holes and three-qubit states -- 3.4. Further developments -- 3.4.1. Microscopic interpretation -- 3.4.2. 4-qubit entanglement and the STU model in D = 3 -- 4. Beyond the STU model -- 4.1. N = 8 supergravity and black holes -- 5. E7 and the tripartite entanglement of seven qubits -- 6. Fano plane entanglement and the octonions -- 6.1. Composition algebras.

6.2. The octonionic tripartite entanglement of seven qubits -- 6.3. Subsectors -- 7. Cubic Jordan algebras and the Freudenthal triple system -- 7.1. Cubic Jordan algebras -- 7.2. The Freudenthal triple system -- 8. The 3-qubit Freudenthal triple system -- 8.1. The FTS rank entanglement classes -- 8.1.1. Rank 1 and the class of separable states -- 8.1.2. Rank 2 and the class of biseparable states -- 8.1.3. Rank 3 and the class of W-states -- 8.1.4. Rank 4 and the class of GHZ-states -- 8.2. SLOCC orbits -- 9. Supersymmetric quantum information -- 10. Supergroups -- 10.1. Grassmann numbers -- 10.2. Super linear algebra -- 10.3. Orthosymplectic superalgebras -- 11. Super Hilbert space and uOSp(1 2) -- 11.0.1. Physical states -- 11.1. The superqubit -- 12. Super entanglement -- 12.1. Two superqubits -- 12.2. Three superqubits -- Acknowledgments -- References -- Weak Value with Decoherence A. Hosoya -- 1. Introduction -- 2. Weak Value -- 3. Weak Measurement with Decoherence -- 3.1. Weak Measurement-Review -- 3.2. Weak Measurement and Environment -- 4. Geometric Phase -- 5. Summary -- Bibliography -- Lectures on Matrix Product Representation of States V. Karimipour and M. Asoudeh -- 1. Introduction -- Part I: Matrix Product States in Quantum Spin Chains -- 2. Quantum Spin Chains -- 3. Ground States of Spin Chains as Matrix Product States -- 3.1. Construction of the MPS -- 3.2. Correlation Functions -- 3.3. Symmetries of the Ground State -- 4. The Parent Hamiltonian -- 4.1. Symmetries of the Hamiltonian -- 4.2. Some Examples -- 4.2.1. The GHZ state -- 4.2.2. The W state -- 4.3. Cluster States -- 4.4. A Simple Spin One-half Chain with Phase Transition -- 4.5. The Majumdar-Ghosh Models -- 4.5.1. The AKLT states -- 5. Conclusion of Part I -- Part II: One Dimensional Reaction Diffusion Systems -- 6. The Master Equation and its Hamiltonian Form.

7. The Stationary State and it's Matrix Product Form -- 8. An Example: The Asymmetric Simple Exclusion Process -- 8.1. Correlation Functions -- 9. The Phase Diagram -- 10. Extensions of the ASEP -- 11. Conclusion of Part II -- 12. Acknowledgements -- References -- On a Possible Definition of Entanglement in Antisymmetric States T. Ichikawa, T. Sasaki, and I. Tsutsui -- Acknowledgements -- References -- Entanglement Measures for Intermediate Separability T. Sasaki, T. Ichikawa, and I. Tsutsui -- 1. Introduction -- 2. Entanglement Measures as Yardsticks of Separability in Arbitrary Subsystems -- 3. Illustrating Rm for some Classes of Globally Entangled States -- 4. Physical Interpretations of the Measures Rm -- 5. Conclusion -- Acknowledgments -- References -- Unruh Effect on Quantum Teleportation and Entanglement: Implications on Black Hole Information K. Shiokawa -- 1. Introduction -- 2. General Formulation -- 3. Entanglement Dynamics of Accelerated Oscillators -- 3.1. Bipartite entanglement -- 3.2. Tripartite entanglement dynamics -- 4. Quantum Teleportation Dynamics with an Accelerated Observer -- 5. Implications -- 6. Acknowledgments -- References -- Systematic Construction of Generalized Bell Inequalities S. Tanimura -- 1. Introduction -- 2. Bell-Clauser-Horne-Shimony-Holt Inequality -- 2.1. Setting up -- 2.2. Hidden-variable theory -- 2.3. Quantum mechanics -- 2.4. Role of locality -- 3. Why Is the BCHSH Inequality Violated? -- 3.1. Interference effect -- 3.2. Noncommutativity and a useful trick -- 4. New Test -- 4.1. New quantity -- 4.2. Hidden-variable theory vs. quantum mechanics -- 5. Concluding Remarks -- Acknowledgements -- References -- On 3-Variable Exponential Polynomials and Quantum Algorithms Y. Ohno, Y. Sasaki, and C. Yamazaki -- 1. Introduction -- 2. The Number of Solutions of Equation.

3. Calculation of the Deterministic Time for a Classical Algorithm -- 4. Calculation of the Time Complexity for a Quantum Algorithm -- 5. Concluding Remarks -- Acknowledgments -- Appendix A. Algebra -- Appendix A.1. Group -- Appendix A.2. Residue class of N -- Appendix A.3. Ring -- Appendix A.4. Field -- References -- Part B Workshop on Matrix Product State Formulation and Density Matrix Renormalization Group Simulations (MPS&DMRG) -- Application of Density Matrix Renormalization Group Method to Photoinduced Phenomena in Strongly Correlated Electron Systems H. Matsueda -- 1. Introduction -- 1.1. Motivation from condensed matter physics side -- 1.2. Quantum information perspective -- 2. Concept of DMRG: A Modern Approach -- 2.1. Superblock and singular value decomposition -- 2.2. Entanglement entropy -- 2.3. Sweeping and matrix product state -- 2.4. Finite-system sweep of DMRG -- 3. Extention of DMRG for Excited States -- 3.1. Targetting excited states into density matrix -- 3.2. Optical spectra -- 3.2.1. Linear optical absorption and optically forbidden spectrum -- 3.2.2. Kramers-Kronig relation -- 3.2.3. Two correction vectors -- 3.3. Angle resolved photoemission spectroscopy -- 3.3.1. DMRG calculation of ARPES spectra -- 3.3.2. Pseudo-momenta -- 4. Time-dependent DMRG -- 4.1. Direct approach similar to dynamical DMRG -- 4.2. MPS and time evolution -- 4.3. Wave function transformation -- 5. Optical Physics of 1D Mott Insulators -- 5.1. Introduction -- 5.2. Fundamental concepts in photoexcited 1D Mott insulators -- 5.2.1. Photo-carrier doping into 1D systems -- 5.2.2. Physics of spin-charge separation, holon-doublon picture -- 5.2.3. Coupling between photocarriers and phonons -- 5.3. Relevant models for optically-excited 1D Mott insulators -- 5.3.1. Hubbard models -- 5.3.2. Holstein and Peierls couplings -- 5.3.3. Pseudo-phonon method.

5.3.4. Holon-doublon model -- 5.4. Photoexcited and single-particle excited states of 1D Mott insulators -- 5.4.1. Degenerate photoexcited states with even and odd parity -- 5.4.2. Effect of on-site and nearest-neighbor Coulomb interactions on Holstein-type electron-phonon coupling -- 5.4.3. Stability of spin-charge separation in the presence of Holstein-type electron-phonon coupling -- 5.5. Ultrafast relaxation dynamics -- 5.5.1. Motivations and purposes -- 5.5.2. Time-dependent quantities -- 5.5.3. Numerical results and perspectives -- 6. What is Effective Spatial Dimension of Fermion-Boson Coupled System? -- 7. Photoinduced States of Multi-Orbital Systems -- 7.1. Manganites -- 7.2. Cobaltites -- 8. Summary -- Acknowledgement -- References -- Density-Matrix Renormalization Group Method for Tomonaga-Luttinger Liquid T. Hikihara -- 1. Introduction -- 2. Combined Scheme of Bosonization and DMRG Methods -- 2.1. Bosonization approach -- 2.2. DMRG results -- 3. Applications -- 3.1. XXZ chain under magnetic field -- 3.2. Two-leg ladder under magnetic field -- 3.3. Zigzag ladder under magnetic field -- 4. Summary -- Acknowledgments -- References -- Supersymmetric Valence-Bond Solid Models - Hidden Order and Dynamics K. Totsuka and K. Hasebe -- 1. Introduction -- 2. Valence Bond Solid Models -- 3. Supersymmetric Valence Bond Solid Models -- 3.1. UOSp(1 2) supersymmetry -- 3.2. UOSp(1 2) SVBS states -- 3.3. Supersymmetric parent Hamiltonian -- 4. Matrix Product Representation -- 4.1. Matrix-product formalism -- 4.2. SMPS formalism -- 5. Magnetic- and Superconducting Properties: "Bosons v.s. Fermions -- 6. Hidden String Order -- 7. Lowlying Excitations -- 8. SU(2 1) SVBS State -- 9. Summary -- References -- Matrix Product States in Quantum Integrable Models H. Katsura and I. Maruyama -- 1. Introduction -- 2. Algebraic Bethe Ansatz for the Spin-1 2 XXZ Chain.

3. Matrix Product State Representation for the Bethe States in the Spin-1 2 XXZ Chain.

This book is a collection of lecture notes/contributions from a summer school on decoherence, entanglement & entropy and a workshop on MPS (matrix product states) & DMRG (density matrix renormalization group). Subjects of the summer school include introduction to MPS, black holes, qubits and octonions, weak measurement, entanglement measures and separability, generalized Bell inequalities, among others. Subjects of the workshop are dedicated to MPS and DMRG. Applications to strongly correlated systems and integrable systems are also mentioned. Contributions to this book are prepared in a self-contained manner so that a reader with a modest background in quantum information and quantum computing may understand the subjects.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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