Quantum theory is one of the most important intellectual developments in the early twentieth century. The confluence of mathematics and quantum physics emerged arguably from Von Neumann's seminal work on the spectral theory of linear operators. This volume arose from a two-month workshop held at the Institute for Mathematical Sciences at the National University of Singapore in July September 2008 on mathematical physics, focusing specifically on operator algebras in quantum theory. This volume is essentially written for graduate students and young researchers so that they can acquire a gentle introduction to the application of operator algebras to quantum information sciences, chaotic and many-body problems. Several lecture notes delivered during the workshop by experts in the field were specially commissioned for this volume.
Author(s): Huzihiro Araki
Series: (Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore
Edition: 1
Publisher: World Scientific Publishing Company
Year: 2010
Language: English
Pages: 221
Tags: Физика;Матметоды и моделирование в физике;
CONTENTS......Page 6
Foreword......Page 8
Preface......Page 10
1. Introduction — Motivations......Page 14
1.1. Motivations......Page 15
1.2. Experimental methods......Page 16
2.1. The free molecule......Page 18
2.2.2. Polarizability interaction......Page 20
2.3.1. Pure state......Page 21
2.3.3. Sudden approximation......Page 22
3.1. Control objectives......Page 23
3.2.1. Target states......Page 25
3.2.4. Optimization with an evolutionary algorithm......Page 28
3.3.1. Target states......Page 31
3.3.2. Strategy of maxima......Page 35
4. Conclusion......Page 36
References......Page 37
Quantum Computing and Devices: A Short Introduction Zhigang Zhang, Viswanath Ramakrishna and Goong Chen......Page 40
2. A Brief Historical Account, Motivation, and Three Fundamental Principles of Quantum Computing......Page 41
2.1. Spins, the Stern-Gerlach experiment and superposition......Page 43
2.2. Entanglement and the Einstein-Podolsky-Rosen (EPR) Problem......Page 46
2.3. Reversibility and irreversibility, the Landauer principle......Page 49
3.1. The electromagnetic field and its quantization......Page 54
3.2. The interaction between a two-level atom and the optical field......Page 57
3.3. Cavity-QED......Page 60
3.4.1. Setting up the fields......Page 62
3.4.2. Storing, sorting, and transporting ultra cold neutral atoms using optical latices......Page 64
3.4.3. Entanglement with ultracold atoms in optical lattices......Page 67
4. Universality of Quantum Gates......Page 70
4.1. The Hamiltonian for the cavity-QED system......Page 71
4.2. The coupled quantum dot Hamiltonian......Page 72
4.3. An explicit conjugation......Page 73
Acknowledgments......Page 75
References......Page 76
1. Introduction......Page 78
2. Hilbert Space Framework of Quantum Dynamics......Page 79
3.1. Koopman-Schr odinger representation......Page 80
3.2. Koopman-Heisenberg representation......Page 83
4. Dynamics of Mixed Classical-Quantum Systems......Page 84
4.1. Example: Stern-Gerlach experiment......Page 85
4.2. Physical interpretation — classical-quantum entanglement......Page 87
5.1. Selection of a polarization subspace......Page 88
5.1.1. Construction of a basis of L K......Page 89
5.1.2. Selection of a polarization subspace L LK by the choice of a subset of the basis......Page 90
5.1.3. De nition of an isomorphism between L LK and Fock space......Page 91
5.2. Toeplitz quantization of the observables......Page 92
5.3. Toeplitz quantization of the generators of the dynamics — geometric quantization......Page 93
5.4. Quantization by coherent states......Page 95
5.4.2. Construction of the coherent states determined by the selection of a polarization subspace......Page 97
5.4.3. Coherent state quantization of observables......Page 99
5.4.4. Coherent state quantization of the generators of the dynamics......Page 100
6. Dequantization by Coherent States......Page 101
6.0.2. (b) Construction of the polarization subspace and of the isomorphism from a given set of unnormalized coherent states......Page 102
6.0.3. (c) Dequantization of the observables — covariant and contravariant symbols......Page 104
7. Conclusions......Page 106
References......Page 107
0. Introduction......Page 110
1. Encoded Qubit......Page 111
2. A Generic Model of Classical Memory......Page 112
3. Davies Generators......Page 114
4. Kitaev Models......Page 117
References......Page 120
1.1. The additivity problem......Page 122
2.1. Additivity and entanglement......Page 124
2.2. Different forms of the additivity property......Page 127
2.3. Nonadditivity of quantum entropy quantities......Page 128
2.4. Infinite-dimensional channels......Page 132
3.1. Canonical commutation relations, symplectic space and complex structures......Page 134
3.2. Gaussian states and channels......Page 137
3.3. The classical capacity......Page 139
References......Page 142
1. Introduction......Page 146
2. Two Qubit Master Equation......Page 147
3. Dissipative Entanglement Generation......Page 149
3.1. Persistence of Entanglement......Page 155
Appendix......Page 156
References......Page 158
1. Introduction......Page 160
2.2. Measurement of observables......Page 162
2.3. Physical meaning of the scattering operator......Page 164
2.5. Alternative kinds of M ller operators......Page 166
3. Scattering Theory for 2-Body Schrodinger Operators......Page 168
3.2. Physical meaning of scattering cross-sections......Page 169
3.3. Long-range case......Page 170
3.4. Freedom of the choice of modi ed M ller operators......Page 171
4.1. Fock spaces......Page 172
4.4. Wick quantization......Page 173
5.1. QFT Hamiltonians......Page 174
5.3. Ground state......Page 175
5.5. The LSZ formalism......Page 176
6.1. Infra-red case A......Page 177
6.3. Infra-red case C......Page 178
6.5. The LSZ formalism for van Hove Hamiltonians......Page 179
7.2. Regular representations of the CCR......Page 180
7.5. Coherent representations......Page 181
7.6. Coherent sectors......Page 182
7.7. Covariant representations......Page 183
7.8. Coherent sectors of a covariant representation......Page 184
8.1. Definition of Pauli-Fierz Hamiltonians......Page 185
8.3. Scattering theory of Pauli-Fierz Hamiltonians......Page 186
8.4. Asymptotic dynamics......Page 188
8.5. Asymptotic completeness......Page 189
8.7. Coherent asymptotic representations......Page 190
Acknowledgments......Page 191
References......Page 192
1. Relative Boundedness and Stability of the First Kind......Page 194
2. Stability of Matter — Stability of the Second Kind......Page 203
3. Hartree-Fock (HF) Theory for Atoms and Molecules......Page 209
4. Bogolubov-Hartree-Fock (BHF) Theory......Page 216
References......Page 220
