Quantum Physics and Geometry (eBook)

Contents 6
Contributors 7
1 Introduction 8
References 11
2 A Very Brief Introduction to Quantum Computing and Quantum Information Theory for Mathematicians 12
2.1 Overview 12
2.2 Quantum Computation as Generalized Probabilistic Computation 13
2.2.1 Classical and Probabilistic Computing via Linear Algebra 13
2.2.2 A Wish List 15
2.2.3 Postulates of Quantum Mechanics and Relevant Linear Algebra 17
2.3 Entanglement Phenomena 19
2.3.1 Super-Dense Coding 19
2.3.2 Quantum Teleportation 20
2.3.3 Bell's Game 21
2.3.3.1 Classical Version 21
2.3.3.2 Quantum Version 21
2.4 Quantum Algorithms 22
2.4.1 Grover's Search Algorithm 22
2.4.2 The Quantum Discrete Fourier Transform 24
2.4.3 The Hidden Subgroup Problem 25
2.5 Classical Information Theory 26
2.5.1 Data Compression: Noiseless Channels 26
2.5.2 Transmission over Noisy Channels 29
2.5.2.1 Capacity of a Noisy Channel 30
2.6 Reformulation of Quantum Mechanics 30
2.6.1 Partial Measurements 30
2.6.2 Mixing Classical and Quantum Probability 31
2.6.3 Reformulation of the Postulates of Quantum Mechanics 33
2.6.4 Expectation and the Uncertainty Principle 33
2.6.5 Pure and Mixed States 34
2.7 Communication Across a Quantum Channel 35
2.8 More on von Neumann Entropy and Its Variants 36
2.9 Entanglement and LOCC 37
2.9.1 LOCC 38
2.9.2 A Partial Order on Probability Distributions Compatible with Entropy 39
2.9.3 A Reduction Theorem 39
2.9.4 Entanglement Distillation (Concentration) and Dilution 40
2.10 Tensor Network States 41
2.11 Representation Theory in Quantum Information Theory 44
2.11.1 Review of Relevant Representation Theory 45
2.11.2 Quantum Marginals and Projections onto Isotypic Subspaces of H d 45
References 47
3 Entanglement, CP-Maps and Quantum Communications 49
3.1 Introduction 49
3.2 Entanglement 54
3.2.1 Quantum Correlations and EPR Paradox 54
3.2.2 Sample of Separability Criteria 56
3.3 Quantum Channels 57
3.3.1 Completely Positive Maps 57
3.3.2 Stinespring Representation 60
3.3.3 Noisy Channels 62
3.4 Quantum Communications 63
3.4.1 Information Processing 64
3.4.2 Relevant No-Go Theorems: Impossible Machines 65
3.4.3 Quantum Teleportation 68
3.4.4 Dense Coding 72
3.5 Final Remarks and Perspectives 74
References 75
4 Frontiers of Open Quantum System Dynamics 77
4.1 Introduction 77
4.2 Open Quantum System Dynamics 78
4.3 Characterization of Dynamics with Memory 80
4.3.1 Generalized Non-Markovianity Measure 84
4.4 Non-Markovian Evolution Equations 85
4.5 Conclusions and Outlook 89
References 89
5 Geometric Constructions over C and F2 for Quantum Information 92
5.1 Introduction 92
5.2 The Geometry of Entanglement 96
5.2.1 Entanglement Under SLOCC, Tensor Rank and Algebraic Geometry 96
5.2.2 The Three-Qubit Classification via Auxiliary Varieties 101
5.2.3 Geometry of Hyperplanes: The Dual Variety 102
5.2.4 Representation Theory and Quantum Systems 106
5.2.5 From Sequence of Simple Lie Algebras to the Classification of Tripartite Quantum Systems with Similar Classes of Entanglement 109
5.3 The Geometry of Contextuality 113
5.3.1 Observable-Based Proofs of Contextuality 113
5.3.2 The Symplectic Polar Space of Rank N and the N-Qubit Pauli Group 114
5.3.3 Geometry of Hyperplanes: Veldkamp Space of a Point-Line Geometry 116
5.3.4 The Finite Geometry of the Two-Qubit and Three-Qubit Pauli Groups and the Hyperplanes of W(2N-1,2) 117
5.3.5 From Commutation Relations of the Three-Qubit Pauli Group to the Weight Diagrams of Simple Lie Algebras 122
5.4 Conclusion 125
References 126
6 Hilbert Functions and Tensor Analysis 130
6.1 Introduction 130
6.2 Tensors and Projective Geometry 131
6.2.1 The Hilbert Function of Finite Sets in Projective Spaces 134
6.3 Results on Tensors from Classical Projective Geometry 141
6.4 Kruskal's Criterion and Terracini's Criterion 146
6.5 A New Result on the Decomposition of Tensors 150
6.6 Final Remarks and Open Problems 153
References 155
7 Differential Geometry of Quantum States, Observablesand Evolution 157
7.1 Introduction 157
7.1.1 On the Many Pictures of Quantum Mechanics 158
7.1.2 Dirac-Schrödinger vs. Heisenberg-Weyl Picture 158
7.1.2.1 Dirac-Schrödinger Picture 159
7.1.2.2 Heisenberg-Born-Jordan 160
7.1.2.3 Other Pictures 160
7.2 A Geometric Picture of Quantum Mechanics 161
7.2.1 Quantum States and Open Systems 169
7.2.1.1 The Qubit 170
7.2.1.2 Open Quantum Systems: the GKLS Equation 172
7.3 Composition of Systems 173
7.3.1 Decomposing a System 173
7.4 Conclusions and Discussion 174
References 175