Korepin Festschrift, The: From Statistical Mechanics To Quantum Information Science - A Collection Of Articles Written In Honor Of The 60th Birthday Of Vladimir Korepin : From Statistical Mechanics to Quantum Information Science

This volume mainly summarizes the invited talks presented at the 5th Asia-Pacific Workshop on Quantum Information Science (APWQIS) in conjunction with a Festschrift in honor of Professor Vladimir Korepin's 60th birthday. In this Festschrift, we have assembled a medley of interesting articles from some of his friends, well-wishers and collaborators. Comprising both reviews of the state-of-the-art and the latest results, this book covers various aspects of quantum information science, including topics like quantum discord, quantum computing, quantum entanglement, etc. CONTENTS 8 Preface 6 Organizing Committee 10 List of Speakers 12 Photos 14 Chapter 1: Cancellation of Ultra-Violet Infinities in One Loop Gravity V. E. Korepin 22 1. Introduction 22 2. Covariant Quantization 23 3. Cancelation of Infinities in one Loop Approximation 25 4. Finite Part of the Generating Functional of the Scattering Matrix 29 5. One Loop Diagram with Two Vertices 30 References 30 Chapter 2: Quantum Discord in a Spin System with Symmetry Breaking B. Tomasello, D. Rossini, A. Hamma and L. Amico 32 1. Introduction 32 2. Quantum, Classical, and Total Correlations 33 3. The XY Model in Transverse Field 34 3.1. The Phase Diagram 35 4. Classical and Quantum Correlations in the Model 36 5. Analysis of Quantum Discord at T = 0 39 6. Quantum Discord at Finite Temperature 41 7. Discussions 46 Acknowledgments 47 References 47 Chapter 3: Entanglement from the Dynamics of an Ideal Bose Gas in a Lattice S. Bose 50 1. Introduction 50 Acknowledgment 57 References 57 Chapter 4: Aspects of the Riemannian Geometry of Quantum Computation H. E. Brandt 58 1. Introduction 58 2. Riemannian Geometry of SU(2n) 59 3. Summary 79 Acknowledgment 79 References 80 Chapter 5: Quantum Mechanics and the Role of Time: Are Quantum Systems Markovian? T. Durt 82 1. Introduction 82 2. Do Quantum Dice Remember? — Experimental Tests 84 2.1. Bohm–Bub’s model and the Papaliolos experiment 85 2.2. Buonomano’s model and the Summhammer experiment 85 2.3. Hidden measurement approach and the Paris–Nord experiment 86 2.3.1. The Poissonian paradigm 87 2.3.2. Departures from the Poissonian paradigm 88 2.3.3. Recent results 89 2.4. Shnoll’s hypotheses and the Brussels experiment 90 3. Sheldrake and Smolin’s Models, and a Related Experimental Proposal 91 3.1. Sheldrake’s model 91 3.2. Smolin’s model 92 3.3. Unfalsifiability of Sheldrake and Smolin’s models — New experimental proposals 93 4. Conclusions 95 Acknowledgments 96 References 96 Chapter 6: Explicit Formula of the Separability Criterion for Continuous Variables Systems K. Fujikawa 98 1. Introduction and Summary 98 2. Details of Analyses 100 2.1. Simon’s criterion 100 2.2. Gaussian states and P-representation 101 2.3. Duan–Giedke–Chirac–Zoller criterion 102 2.4. Hierarchy of separability criterions 103 3. Discussion and Related References 103 References 103 Chapter 7: Yang–Baxter Equations in Quantum Information M.-L. Ge and K. Xue 106 1. Introduction 106 2. Two Types of Braiding Matrices. Yang–Baxter Equation and Temperley–Lieb Algebra 107 3. 2-D Topological Basis of Braiding Matrices 110 4. Unified Form for Both Type-1 and Type-2 BRM’s 112 5. Two Types of BRM in Physics and Extremization of l1-Norm 115 5.1. Motivation of l1-norm involution 115 5.2. Extremization of D-function and l1-norm 115 6. Bogoliubov’s Hamiltonian Induced from Dirac’s Hamiltonian via Braid Matrix 116 7. Optical Simulation of YBE 118 8. Topological Basis for BWA 119 8.1. Matrix form of E12, E23, S12 and S23 for S† = S 121 8.2. Application to the “standard” q-deformed spin-1 model 122 9. Conclusion 123 Acknowledgments 123 References 124 Chapter 8: Nondistillable Entanglement Guarantees Distillable Entanglement L. Chen and M. Hayashi 126 1. Introduction 126 2. Unification of Entanglement Criterion 129 3. Classification with Reduced States 131 4. Generalization to Multipartite System 134 5. Conclusions 136 6. Acknowledgments and Appendices 136 Appendix 136 References 137 Chapter 9: Reduced Density Matrix and Entanglement Entropy of Permutationally Invariant Quantum Many-Body Systems V. Popkov and M. Salerno 140 1. Introduction 140 2. Model Equation and Basic Definitions 142 3. RDM Properties and Block Diagonal Form 144 4. Analytical Expression of RDM Elements 148 5. Spectral Properties of RDM and Entanglement Entropy 149 6. Moments of the Reduced Density Matrix and Renyi and Tsallis Entropies 152 7. Conclusions 155 Acknowledgments 155 Appendix A. RDM Elements for Symmetric States 156 Appendix B. RDM Elements in the Thermodynamic Limit 156 References 160 Chapter 10: Solitons Experience for Black Hole Production in Ultrarelativistic Particle Collisions I. Ya. Aref ’eva 162 1. Introduction 162 2. Transition Amplitudes and Cross Section of the BH/WH Production 163 3. Ultrarelativistic Particles on the Brane 165 3.1. D-dimensional Planckian energy 165 3.2. D-dimensional gravitational model of relativistic particles 166 4. Shock Wave as a Model of Ultra Relativistic Moving Particle 167 5. D-Dimensional Thorne Hoop Conjecture and Geometrical Cross Section 169 6. Looking from the Rest Frame of One of the Incident Particles 171 7. BH Formation and the Eikonal Approximation 172 8. Transition Amplitudes and Cross Section for Higher Dimensional Gravity and Matter Living on the Brane 174 Acknowledgments 175 Appendix A. BH as an Initial/Final Data 175 A.1. (Weakly) asymptotically simple space–time 176 A.2. Asymptotically flat space–time 177 A.3. Global (partial) Cauchy surface 177 A.4. Causal future J+(p) and causal past J-(p) of the point 178 A.5. BHs in asymptotically flat space–time 178 A.6. Future (strongly) asymptotically predictable space–time 178 A.7. BH on a surface 178 A.8. Boundary conditions in the path integral representation 179 References 179 Chapter 11: Sine–Gordon Theory in the Repulsive Regime, Thermodynamic Bethe Ansatz and Minimal Models H. Itoyama 182 1. Introduction 182 References 189 Chapter 12: On Some Algebraic and Combinatorial Properties of Dunkl Elements A. N. Kirillov 192 1. Introduction 192 2. Dunkl Elements 197 3. Algebra 3HTn 200 4. Combinatorics of Associative Quasi-Classical Yang–Baxter Algebras 207 4.1. Combinatorics of Coxeter element 208 4.2. Grothendieck and q-Schroderpolynomials 210 4.2.1. Schroder paths and polynomials 210 4.2.2. Principal specialization of Grothendieck polynomials and q-Schroder polynomials 214 4.3. The “longest element” and Chan–Robbins polytope 216 Acknowledgments 218 References 218 Chapter 13: Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits M. Saniga 220 Acknowledgments 222 References 222 Chapter 14: Monogamy of Entanglement, N-Representability Problems and Ground States T.-C. Wei 224 1. Introduction 224 2. Mean-Field Approximation 225 3. N-Representability Problem 226 3.1. 3-representable 226 3.2. n-representable 228 3.3. Ground-state entanglement 228 4. Concluding Remarks 229 Acknowledgment 230 References 230 Scientific Programs 232 List of Participants 236 Preface; CONTENTS; Organizing Committee; List of Speakers; Photos; Chapter 1: Cancellation of Ultra-Violet Infinities in One Loop Gravity V.E. Korepin; 1. Introduction; 2. Covariant Quantization; 3. Cancelation of Infinities in one Loop Approximation; 4. Finite Part of the Generating Functional of the Scattering Matrix; 5. One Loop Diagram with Two Vertices; References; Chapter 2: Quantum Discord in a Spin System with Symmetry Breaking B. Tomasello, D. Rossini, A. Hamma and L. Amico; 1. Introduction; 2. Quantum, Classical, and Total Correlations; 3. The XY Model in Transverse Field