Mathematical Horizons for Quantum Physics

The following sections are included:

• CONTENTS

• FOREWORD

• PREFACE

We review the motivations and recent theoretical and experimental progress in laser alignment or orientation of molecules in space. We present a method to produce field free aligned or oriented linear molecules using a succession of short laser pulses. The problem of orienting molecules is an example of a unitary control problem in a Hilbert space with infinite dimension. The originality of our method relies on a precise construction of oriented quantum target states which maximize the orientation in finite dimensional subspaces. Orientation at zero temperature (pure state control) and of molecular thermal ensemble (mixed state control) are investigated.

Quantum computation is a new interdisciplinary research enterprise involving physicists, mathematicians, computer scientists and engineers. In this chapter, we give a quick introduction of the subject. Three fundamental principles of quantum informatics: superposition, entanglement and reversibility, are described first. The quantum devices of cavity-QED and optical lattices, as candidates for quantum computing gates are introduced next. Finally, we offer an algebraic treatment by conjugacy to obtain some results on universal quantum gates by explicit conjugation.

We describe quantum and classical Hamiltonian dynamics in a common Hilbert space framework, that allows the treatment of mixed quantum-classical systems. The analysis of some examples illustrates the possibility of entanglement between classical and quantum systems. We give a summary of the main tools of Berezin-Toeplitz and geometric quantization, that provide a relation between the classical and the quantum models, based essentially on the selection of a subspace of the classical Hilbert space. Coherent states provide a systematic tool for the inverse process, called dequantization, that associates a classical Hamiltonian system to a given quantum dynamics through the choice of a complete set of coherent states.

Promising candidates for quantum memory are N-spin systems with specially designed Hamiltonians weakly coupled to heat baths. Their dynamics can be described by quantum dynamical semigroups of the Davies type. The rigorous results concerning 2D and 4D Kitaev models are briey reviewed and their physical meaning is discussed.

We survey the modern status of two major analytical problems in quantum information theory. One is the famous additivity conjecture for information quantities related to the classical capacity of quantum channel. Another is less familiar and less studied problem of optimizers of the information quantities characterizing Bosonic Gaussian channels. We show that for gauge-invariant channels, the validity of the Gaussian minimizer conjecture for the output entropy implies Gaussianity of the ensemble maximizing the χ-capacity.

Two non-directly interacting qubits with equal oscillation frequencies can be entangled on a short time-scale by a purely dissipative mechanism, that is, by the irreversible reduced dynamics that result from weakly coupling them to a common suitably engineered environment. Depending on the two-point time-correlation functions of the latter, the generated entanglement may also persist asymptotically in time.

The lecture notes are devoted to some topics in scattering theory for certain models inspired by quantum field theory. As a toy example, we describe scattering theory for van Hove Hamiltonians, where all basic objects can be computed exactly. We also sketch the formalism, basic results and some open problems about the so-called Pauli-Fierz Hamiltonians — a class of models describing a small quantum system interacting with a bosonic field, which have an interesting and nontrivial scattering theory.

In these notes, we present some key aspects of the mathematical description of the quantum mechanics of nonrelativistic matter, i.e., atoms and molecules. Special focus lies on the concepts of infinitesimal perturbations, stability of matter, the Hartree-Fock approximation and its justification, and its generalization to a variation over quasifree states which is termed Bogolubov-Hartree-Fock theory.