INTRODUCTION TO THE SCIENCE OF DYNAMICS

By G. LUMIY

CONTENTS QUANTUM MECHANICS STATES AND OPERATORS STATE OF A QUANTUM SYSTEM LINEAR OPERATORS STATE OF COMPOSITE SYSTEMS AND TENSOR PRODUCT SPACES STATE OF AN ENSEMBLE; DENSITY OPERATOR VECTOR AND MATRIX REPRESENTATION OF STATES AND OPERA- TORS OBSERVABLES AND MEASUREMENT OBSERVABLES THE MEASUREMENT POSTULATE MEASUREMENTS ON ENSEMBLES DYNAMICS OF QUANTUM SYSTEMS SCHR¨ODINGER PICTURE HEISENBERG AND INTERACTION PICTURE NOTES INTERPRETATION OF QUANTUM DYNAMICS AS INFORMATION PRO- CESSING DIRECT SUM VERSUS TENSOR PRODUCT FOR COMPOSITE SYSTEMS EXERCISES MODELING OF QUANTUM CONTROL SYSTEMS; EXAMPLES QUANTUM THEORY OF INTERACTION OF PARTICLES AND FIELD CLASSICAL ELECTRODYNAMICS CANONICAL QUANTIZATION AN EXAMPLE OF CANONICAL QUANTIZATION: THE QUANTUM HARMONIC OSCILLATOR QUANTUM MECHANICAL HAMILTONIAN APPROXIMATIONS AND MODELING; MOLECULAR SYSTEMS APPROXIMATIONS FOR MOLECULAR AND ATOMIC SYSTEMS CONTROLLED SCHR¨ODINGER WAVE EQUATION SPIN DYNAMICS AND CONTROL INTRODUCTION OF THE SPIN DEGREE OF FREEDOM IN THE DY- NAMICS OF MATTER AND ?ELDS SPIN NETWORKS AS CONTROL SYSTEMS MATHEMATICAL STRUCTURE OF QUANTUM CONTROL SYSTEMS NOTES AND REFERENCES CONTROLLABILITY LIE ALGEBRAS AND LIE GROUPS BASIC DE?NITIONS FOR LIE ALGEBRAS LIE GROUPS CONTROLLABILITY TEST: THE DYNAMICAL LIE ALGEBRA PROCEDURE TO GENERATE A BASIS OF THE DYNAMICAL LIE AL- GEBRA UNIFORM ?NITE GENERATION OF COMPACT LIE GROUPS AND UNIVERSAL QUANTUM GATES NOTIONS OF CONTROLLABILITY FOR THE STATE PURE STATE CONTROLLABILITY LIE TRANSFORMATION GROUPS COSET SPACES AND HOMOGENEOUS SPACES THE SPECIAL UNITARY GROUP AND ITS ACTION ON THE UNIT SPHERE THE SYMPLECTIC GROUP AND ITS ACTION ON THE UNIT SPHERE TEST FOR PURE STATE CONTROLLABILITY EQUIVALENT STATE CONTROLLABILITY EQUALITY OF ORBITS DENSITY MATRIX CONTROLLABILITY ALTERNATE TESTS OF CONTROLLABILITY PURE STATE CONTROLLABILITY AND EXISTENCE OF CONSTANTS OF MOTION BIBLIOGRAPHICAL NOTES SOME OPEN PROBLEMS OBSERVABILITY AND STATE DETERMINATION QUANTUM STATE TOMOGRAPHY EXAMPLE: QUANTUM TOMOGRAPHY OF A SPIN 2 PARTICLE GENERAL QUANTUM TOMOGRAPHY 2 PARTICLE (CTD) OBSERVABILITY EQUIVALENCE CLASSES OF INDISTINGUISHABLE STATES; PARTI- TION OF THE STATE SPACE OBSERVABILITY AND METHODS FOR STATE RECONSTRUCTION OBSERVABILITY CONDITIONS AND TOMOGRAPHIC METHODS SYSTEM THEORETIC METHODS FOR QUANTUM STATE RECONSTRUC- TION LIE GROUP DECOMPOSITIONS AND CONTROL DECOMPOSITIONS OF SU(2) AND CONTROL OF TWO LEVEL SYSTEMS THE LIE GROUPS SU(2) AND SO(3) EULER DECOMPOSITION OF SU(2) AND SO(3) DETERMINATION OF THE ANGLES IN THE EULER DECOMPOSITION OF SU(2) APPLICATION TO THE CONTROL OF TWO LEVEL QUANTUM SYSTEMS DECOMPOSITION IN PLANAR ROTATIONS CARTAN DECOMPOSITIONS CARTAN DECOMPOSITION OF SEMISIMPLE LIE ALGEBRAS THE DECOMPOSITION THEOREM FOR LIE GROUPS RE?NEMENT OF THE DECOMPOSITION; CARTAN SUBALGEBRAS CARTAN DECOMPOSITIONS OF SU(N) CARTAN INVOLUTIONS OF SU(N) AND QUANTUM SYMMETRIES COMPUTATION OF THE FACTORS IN THE CARTAN DECOMPOSI- TIONS OF SU(N) LEVI DECOMPOSITION IDEALS AND NORMAL SUBGROUPS SOLVABLE LIE ALGEBRAS EXAMPLES OF APPLICATION OF DECOMPOSITIONS TO CONTROL CONTROL OF TWO COUPLED SPIN 2 PARTICLES WITH ISING INTER- ACTION 2 PARTICLES WITH HEISENBERG INTERACTION OPTIMAL CONTROL OF QUANTUM SYSTEMS FORMULATION OF THE OPTIMAL CONTROL PROBLEM OPTIMAL CONTROL PROBLEMS OF MAYER, LAGRANGE AND BOLZA OPTIMAL CONTROL PROBLEMS FOR QUANTUM SYSTEMS THE NECESSARY CONDITIONS OF OPTIMALITY GENERAL NECESSARY CONDITIONS OF OPTIMALITY THE NECESSARY OPTIMALITY CONDITIONS FOR QUANTUM CON- TROL PROBLEMS EXAMPLE: OPTIMAL CONTROL OF A TWO LEVEL QUANTUM SYSTEM TIME OPTIMAL CONTROL OF QUANTUM SYSTEMS THE TIME OPTIMAL CONTROL PROBLEM; BOUNDED CONTROL MINIMUM TIME CONTROL WITH UNBOUNDED CONTROL; RIE- MANNIAN SYMMETRIC SPACES NUMERICAL METHODS FOR OPTIMAL CONTROL OF QUANTUM SYSTEMS METHODS USING DISCRETIZATION ITERATIVE METHODS NUMERICAL METHODS FOR TWO POINTS BOUNDARY VALUE PROB- LEMS MORE TOOLS FOR QUANTUM CONTROL SELECTIVE POPULATION TRANSFER VIA FREQUENCY TUNING TIME DEPENDENT PERTURBATION THEORY ADIABATIC CONTROL STIRAP LYAPUNOV CONTROL OF QUANTUM SYSTEMS QUANTUM CONTROL PROBLEMS IN TERMS OF A LYAPUNOV FUNC- TION DETERMINATION OF THE CONTROL FUNCTION STUDY OF THE ASYMPTOTIC BEHAVIOR OF THE STATE ? ANALYSIS OF QUANTUM EVOLUTIONS; ENTANGLEMENT, ENTANGLE- MENT MEASURES AND DYNAMICS ENTANGLEMENT OF QUANTUM SYSTEMS BASIC DE?NITIONS AND NOTIONS TESTS OF ENTANGLEMENT MEASURES OF ENTANGLEMENT AND CONCURRENCE DYNAMICS OF ENTANGLEMENT THE TWO QUBITS EXAMPLE THE ODD-EVEN DECOMPOSITION AND CONCURRENCE DYNAMICS RECURSIVE DECOMPOSITION OF DYNAMICS IN ENTANGLING AND LOCAL PARTS LOCAL EQUIVALENCE OF STATES GENERAL CONSIDERATIONS ON DIMENSIONS INVARIANTS AND POLYNOMIAL INVARIANTS SOME SOLVED CASES APPLICATIONS OF QUANTUM CONTROL AND DYNAMICS NUCLEAR MAGNETIC RESONANCE EXPERIMENTS BASICS OF NMR 2-DIMENSIONAL NMR CONTROL PROBLEMS IN NMR MOLECULAR SYSTEMS CONTROL PULSE SHAPING OBJECTIVES AND TECHNIQUES OF MOLECULAR CONTROL ATOMIC SYSTEMS CONTROL; IMPLEMENTATIONS OF QUANTUM INFOR- MATION PROCESSING WITH ION TRAPS PHYSICAL SET-UP OF THE TRAPPED IONS QUANTUM INFORMA- TION PROCESSOR CLASSICAL HAMILTONIAN PRACTICAL IMPLEMENTATION OF DI?ERENT INTERACTION HAMIL- TONIANS THE CONTROL PROBLEM: SWITCHING BETWEEN HAMILTONIANS INDEX CHAPTER STATES AND HILBERT SPACES THE STATE OF A QUANTUM MECHANICAL SYSTEM IS REPRESENTED BY A VECTOR IN A SEPARABLE HILBERT SPACE THE WORD ‘REPRESENTED’ MEANS HERE THAT KNOWLEDGE OF THE STATE VECTOR GIVES COMPLETE INFORMATION ON THE PROPERTIES OF THE SYSTEM A HILBERT SPACE H (SEE, EG, [179] PG 76 ?) IS A COMPLEX VECTOR SPACE WITH AN INNER PRODUCT THAT IS AN OPERATION (·, ·) : H × H ?CI WHICH SATIS?ES THE FOLLOWING PROPERTIES 1 (?Y, ?X) = (?X, ?Y)?, (11) WHERE A?DENOTES THE COMPLEX CONJUGATED OF A, 2 (?X + ?Y,?Z) = (?X,?Z) + (?Y,?Z), 3 (??X, ?Y) = ??(?X, ?Y), (12) FOR ANY COMPLEX SCALAR ?, AND, 4 (?X, ?X) ?0,