1 Quantum Mechanics .......................................... 1
1.1 States and Operators ....................................... 1
1.1.1 State of a quantum system ........................... 1
1.1.2 Linear operators .................................... 5
1.1.3 State of composite systems and tensor product
spaces .............................................. 8
1.1.4 State of an ensemble; Density operator ............. 11
1.1.5 Vector and matrix representation of states and
operators .......................................... 13
1.2 Observables and Measurement ............................... 15
1.2.1 Observables ........................................ 15
1.2.2 The measurement postulate .......................... 20
1.2.3 Measurements on ensembles .......................... 24
1.3 Dynamics of Quantum Systems ............................... 25
1.3.1 Schrцdinger picture ................................ 26
1.3.2 Heisenberg and interaction picture ................. 29
1.4 Notes ..................................................... 30
1.4.1 Interpretation of quantum dynamics as information
processing ......................................... 30
1.4.2 Direct sum versus tensor product for composite
systems ............................................ 31
1.5 Exercises ................................................. 32
2 Modeling of Quantum Control Systems; Examples ............. 35
2.1 Quantum Theory of Interaction of Particles and Field ...... 35
2.1.1 Classical electrodynamics .......................... 36
2.1.2 Canonical quantization ............................. 46
2.1.3 An example of canonical quantization: The quantum
harmonic oscillator ................................ 50
2.1.4 Quantum mechanical Hamiltonian ..................... 53
2.2 Approximations and Modeling; Molecular Systems ............ 56
2.2.1 Approximations for molecular and atomic systems .... 56
2.2.2 Controlled Schrцdinger wave equation ............... 59
2.3 Spin Dynamics and Control ................................. 62
2.3.1 Introduction of the spin degree of freedom in the
dynamics of matter and fields ...................... 63
2.3.2 Spin networks as control systems ................... 66
2.4 Mathematical Structure of Quantum Control Systems ......... 69
2.5 Notes and References ...................................... 72
2.6 Exercises ................................................. 73
3 Controllability ........................................... 75
3.1 Lie Algebras and Lie Groups ............................... 76
3.1.1 Basic definitions for Lie algebras ................. 76
3.1.2 Lie groups ......................................... 79
3.2 Controllability Test: The Dynamical Lie Algebra ........... 81
3.2.1 Procedure to generate a basis of the dynamical
Lie algebra ........................................ 82
3.2.2 Uniform finite generation of compact Lie groups
and universal quantum gates ........................ 83
3.3 Notions of Controllability for the State .................. 84
3.4 Pure State Controllability ................................ 85
3.4.1 Lie transformation groups .......................... 86
3.4.2 Coset spaces and homogeneous spaces ................ 87
3.4.3 The special unitary group and its action on the
unit sphere ........................................ 88
3.4.4 The symplectic group and its action on the unit
sphere ............................................. 89
3.4.5 Test for pure state controllability ................ 93
3.5 Equivalent State Controllability .......................... 94
3.6 Equality of Orbits ........................................ 95
3.6.1 Density matrix controllability ..................... 97
3.7 Notes and References ...................................... 98
3.7.1 Alternate tests of controllability ................. 98
3.7.2 Pure state controllability and existence of
constants of motion ............................... 100
3.7.3 Bibliographical notes ............................. 102
3.7.4 Some open problems ................................ 102
3.8 Exercises ................................................ 103
4 Observability and State Determination .................... 107
4.1 Quantum State Tomography ................................. 107
4.1.1 Example: Quantum tomography of a spin-1/2
particle .......................................... 107
4.1.2 General quantum tomography ........................ 109
4.1.3 Example: Quantum tomography of a spin-1/2 particle
(ctd.) ............................................ 111
4.2 Observability ............................................ 113
4.2.1 Equivalence classes of indistinguishable states;
Partition of the state space ...................... 114
4.3 Observability and Methods for State Reconstruction ....... 118
4.3.1 Observability conditions and tomographic methods .. 118
4.3.2 System theoretic methods for quantum state
reconstruction .................................... 119
4.4 Notes and References ..................................... 121
4.5 Exercises ................................................ 121
5 Lie Group Decompositions and Control ..................... 123
5.1 Decompositions of SU(2) and Control of Two Level
Systems .................................................. 125
5.1.1 The Lie groups SU(2) and SO(3) .................... 125
5.1.2 Euler decomposition of SU(2) and SO(3) ............ 126
5.1.3 Determination of the angles in the Euler
decomposition of SU(2) ............................ 127
5.1.4 Application to the control of two level quantum
systems ........................................... 128
5.2 Decomposition in Planar Rotations ........................ 130
5.3 Cartan Decompositions .................................... 131
5.3.1 Cartan decomposition of semisimple Lie algebras ... 132
5.3.2 The decomposition theorem for Lie groups .......... 132
5.3.3 Refinement of the decomposition; Cartan
subalgebras ....................................... 133
5.3.4 Cartan decompositions of su(n) .................... 135
5.3.5 Cartan involutions of su(n) and quantum
symmetries ........................................ 137
5.3.6 Computation of the factors in the Cartan
decompositions of SU(n) ........................... 139
5.4 Levi Decomposition ....................................... 145
5.4.1 Ideals and normal subgroups ....................... 145
5.4.2 Solvable Lie algebras ............................. 146
5.4.3 Levi decomposition ................................ 147
5.5 Examples of Application of Decompositions to Control ..... 147
5.5.1 Control of two coupled spin-1/2 particles with
Ising interaction ................................. 148
5.5.2 Control of two coupled spin-1/2 particles with
Heisenberg interaction ............................ 150
5.6 Notes and References ..................................... 153
5.7 Exercises ................................................ 154
6 Optimal Control of Quantum Systems ....................... 157
6.1 Formulation of the Optimal Control Problem ............... 158
6.1.1 Optimal control problems of Mayer, Lagrange and
Bolza ............................................. 158
6.1.2 Optimal control problems for quantum systems ...... 160
6.2 The Necessary Conditions of Optimality ................... 162
6.2.1 General necessary conditions of optimality ........ 162
6.2.2 The necessary optimality conditions for quantum
control problems .................................. 166
6.3 Example: Optimal Control of a Two Level Quantum System ... 166
6.4 Time Optimal Control of Quantum Systems .................. 169
6.4.1 The time optimal control problem; Bounded
control ........................................... 171
6.4.2 Minimum time control with unbounded control;
Riemannian symmetric spaces ....................... 175
6.5 Numerical Methods for Optimal Control of Quantum
Systems .................................................. 182
6.5.1 Methods using discretization ...................... 183
6.5.2 Iterative methods ................................. 183
6.5.3 Numerical methods for two points boundary value
problems .......................................... 186
6.6 Notes and References ..................................... 187
6.7 Exercises ................................................ 188
7 More Tools for Quantum Control ........................... 191
7.1 Selective Population Transfer via Frequency Tuning ....... 191
7.2 Time Dependent Perturbation Theory ....................... 196
7.3 Adiabatic Control ........................................ 198
7.4 STIRAP ................................................... 201
7.5 Lyapunov Control of Quantum Systems ...................... 205
7.5.1 Quantum control problems in terms of a Lyapunov
function .......................................... 205
7.5.2 Determination of the control function ............. 208
7.5.3 Study of the asymptotic behavior of the state p ... 208
7.6 Notes and References ..................................... 214
7.7 Exercises ................................................ 215
8 Analysis of Quantum Evolutions; Entanglement,
Entanglement Measures and Dynamics ....................... 217
8.1 Entanglement of Quantum Systems .......................... 218
8.1.1 Basic definitions and notions ..................... 218
8.1.2 Tests of entanglement ............................. 223
8.1.3 Measures of entanglement and concurrence .......... 231
8.2 Dynamics of Entanglement ................................. 238
8.2.1 The two qubits example ............................ 240
8.2.2 The odd-even decomposition and concurrence
dynamics .......................................... 243
8.2.3 Recursive decomposition of dynamics in
entangling and local parts ........................ 248
8.3 Local Equivalence of States .............................. 251
8.3.1 General considerations on dimensions .............. 252
8.3.2 Invariants and polynomial invariants .............. 255
8.3.3 Some solved cases ................................. 257
8.4 Notes and References ..................................... 257
8.5 Exercises ................................................ 259
9 Applications of Quantum Control and Dynamics ............. 261
9.1 Nuclear Magnetic Resonance Experiments ................... 261
9.1.1 Basics of NMR ..................................... 261
9.1.2 2-Dimensional NMR ................................. 266
9.1.3 Control problems in NMR ........................... 268
9.2 Molecular Systems Control ................................ 269
9.2.1 Pulse shaping ..................................... 269
9.2.2 Objectives and techniques of molecular control .... 270
9.3 Atomic Systems Control; Implementations of Quantum
Information Processing with Ion Traps .................... 272
9.3.1 Physical set-up of the trapped ions quantum
information processor ............................. 273
9.3.2 Classical Hamiltonian ............................. 274
9.3.3 Quantum mechanical Hamiltonian .................... 275
9.3.4 Practical implementation of different
interaction Hamiltonians .......................... 277
9.3.5 The control problem: Switching between
Hamiltonians ...................................... 282
9.4 Notes and References ..................................... 282
9.5 Exercises ................................................ 283
A Positive and Completely Positive Maps, Quantum
Operations and Generalized Measurement Theory ............ 287
A.l Positive and Completely Positive Maps .................... 287
A.2 Quantum Operations and Operator Sum Representation ....... 288
A.3 Generalized Measurement Theory ........................... 289
В Lagrangian and Hamiltonian Formalism in Classical
Electrodynamics .......................................... 291
B.l Lagrangian Mechanics ..................................... 291
B.2 Extension of Lagrangian Mechanics to Systems with
Infinite Degrees of Freedom .............................. 296
B.3 Lagrangian and Hamiltonian Mechanics for a System of
Interacting Particles and Field .......................... 299
С Cartan Semisimplicity Criterion and Calculation of the
Levi Decomposition ....................................... 305
C.l The Adjoint Representation ............................... 305
C.2 Cartan Semisimplicity Criterion .......................... 306
C.3 Quotient Lie Algebras .................................... 306
C.4 Calculation of the Levi Subalgebra in the Levi
Decomposition ............................................ 307
C.5 Algorithm for the Levi Decomposition ..................... 307
D Proof of the Controllability Test of Theorem 3.2.1 ....... 309
E The Baker-Campbell-Hausdorff Formula and Some
Exponential Formulas ..................................... 315
F Proof of Theorem 6.2.1 ................................... 317
References .................................................... 321
Index ......................................................... 337
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