Entanglement and Decoherence: Foundations and Modern Trends Book

1 Geometry of State Spaces A. Uhlmann B. Crell 1

1.1 Introduction 1

1.2 Geometry of Pure States 4

1.2.1 Norm and Distance in Hilbert Space 4

1.2.2 Length of Curves in H 4

1.2.3 Distance and Length 6

1.2.4 Curves on the Unit Sphere 7

1.2.5 Phases 9

1.2.6 Fubini-Study Distance 10

1.2.7 Fubini-Study Metric 12

1.2.8 Symmetries 13

1.2.9 Comparison with Other Norms 15

1.3 Operators, Observables, and States 16

1.3.1 States and Expectation Values 17

1.3.2 Subalgebras and Subsystems 19

1.3.3 Classification of Finite Quantum Systems 23

1.3.4 All Subsystems for dim H1.4 Transition Probability, Fidelity, and Bures Distance 35

1.4.1 Purification 36

1.4.2 Transition Probability and Fidelity 37

1.4.3 Optimization 38

1.4.4 Why the Bures Distance Is a Distance 40

1.4.5 Expressions for Fidelity and Transition Probability 43

1.4.6 Estimates and a "Hidden Symmetry" 47

1.4.7 "Operational Fidelity" 49

1.5 Appendix: The Geometrical Mean 50

1.5.1 Geometric Mean and Fidelity 52

1.5.2 The Transformer Identity 53

1.5.3 #-Convexity 57

References 58

2 Basic Concepts of Entangled States F. Mintert C. Viviescas A. Buchleitner 61

2.1 Introduction 61

2.2 Entangled States 61

2.2.1 Pure States 62

2.2.2 Mixed States 63

2.3 Separability Criteria 64

2.3.1 Pure States 64

2.3.2 Mixed States 66

2.4 Entanglement Monotones and Measures 71

2.4.1 General Considerations 71

2.4.2 Some Specific Monotones and Measures 78

References 86

3 Topology and Quantum Computing L.H. Kauffman S.J. Lomonaco 87

3.1 Introduction 87

3.2 Knots and Braids 90

3.3 Quantum Mechanics and Quantum Computation 94

3.3.1 What Is a Quantum Computer?95

3.4 Braiding Operators and Universal Quantum Gates 96

3.4.1 Universal Gates 99

3.5 A Remark About EPR, Entanglement and Bell's Inequality 102

3.6 The Aravind Hypothesis 104

3.7 SU(2) Representations of the Artin Braid Group 104

3.8 The Bracket Polynomial and the Jones Polynomial 110

3.8.1 The State Summation 113

3.8.2 Quantum Computation of the Jones Polynomial 114

3.8.3 The Hadamard Test 117

3.9 Quantum Topology, Cobordism Categories, Temperley-Lieb Algebra and Topological Quantum Field Theory 118

3.10 Braiding and Topological Quantum Field Theory 127

3.11 Spin Networks and Temperley-Lieb Recoupling Theory 135

3.11.1 Evaluations 140

3.11.2 Symmetry and Unitarity 141

3.12 Fibonacci Particles 145

3.13 The Fibonacci Recoupling Model 148

3.14 Quantum Computation of Colored Jones Polynomials and the Witten-Reshetikhin-Turaev Invariant 150

References 153

4 Entanglement in Phase Space A.M. Ozorio de Almeida 157

4.1 Introduction 157

4.2 Entanglement and Classical Physics 160

4.3 Classical-Quantum Correspondence 163

4.4 Semiclassical Quantum States 169

4.5 Operator Representations and Double Phase Space 175

4.6 The Wigner Function and the Chord Function 182

4.7 The Partial Trace: Sections and Projections 193

4.8 Generating a Classical Entanglement: The EPR State 200

4.9 Entanglement and Decoherence 206

4.10 A Semiclassical Picture of Entanglement 208

References 215

5 Introduction to Decoherence Theory K. Hornberger 221

5.1 The Concept of Decoherence 221

5.1.1 Decoherence in a Nutshell 222

5.1.2 General Scattering Interaction 224

5.1.3 Decoherence as an Environmental Monitoring Process 225

5.1.4 A Few Words on Nomenclature 229

5.2 Case Study: Dephasing of Qubits 230

5.2.1 An Exactly Solvable Model 230

5.2.2 The Continuum Limit 234

5.2.3 Dephasing of N Qubits 237

5.3 Markovian Dynamics of Open Quantum Systems 239

5.3.1 Quantum Dynamical Semigroups 240

5.3.2 The Lindblad Form 242

5.3.3 Quantum Trajectories 245

5.3.4 Exemplary Master Equations 249

5.4 Microscopic Derivations 255

5.4.1 The Weak Coupling Formulation 255

5.4.2 The Monitoring Approach 259

5.4.3 Collisional Decoherence of a Brownian Particle 262

5.4.4 Decoherence of a Quantum Dot 267

5.5 Robust States and the Pointer Basis 270

5.5.1 Nonlinear Equation for Robust States 271

5.5.2 Applications 273

References 274

6 Diffusive Spin Transport C.A. Müet;ller 277

6.1 Introduction 277

6.2 Spin Relaxation 278

6.2.1 Spin - A Primer on Rotations 279

6.2.2 Master Equation Approach to Spin Relaxation 282

6.2.3 Irreducible Scalar Spin Relaxation Rates 287

6.3 Diffusion 294

6.3.1 Transport 294

6.3.2 Momentum - a Primer on Translations 295

6.3.3 Master Equation Approach to Diffusion 296

6.3.4 Linear Response and Diffusion Constant 303

6.4 Diffusive Spin Transport 306

6.4.1 Master Equation Approach to Diffusive Spin Transport 306

6.4.2 Quantum Corrections 309

References 312

Index 315