PhD lectures from the 2023/24 series
November 8th, 2023
Ángela Capel Cuevas (Universität Tübingen)
Quantum entropy and trace inequalities
In this talk, I will give an elementary introduction to the subject of quantum entropies and trace inequalities, with a special focus on results that are relevant to quantum information theory. First, we will discuss various different notions of quantum entropies that extend those of classical entropies, and we will show several of their fundamental properties, in particular under the application of quantum channels. We will put a special focus on those properties that differ from their classical counterparts. Next, we will describe some contexts within quantum Shannon theory in which the use of quantum entropies is fundamental, such as for quantum hypothesis testing or to estimate quantum channel capacities.
Lecture Notes - Lecture Notes (highlighted)
December 13th, 2023
Stefano Marcantoni (Université Côte d'Azur, LJAD)
Introduction to the theory of open quantum systems
In this lecture, I will introduce the framework commonly used to describe the dynamics of open quantum systems. This is based on completely positive dynamical maps and markovian master equations in GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) form. I will discuss the general structure of the latter for an N-level system and present some simple examples. Among them, I will focus on the so-called "Davies generator" induced by the interaction with a thermal Bose field, whose rigorous derivation was originally given by Davies in the van Hove scaling limit and has been recently improved by Merkli uniformly in time. Finally, I will also mention a few directions of current research and point to some relevant literature.
February 21st, 2024
David Mitrouskas (ISTA Austria)
A brief introduction to translation-invariant particle-field models
In this lecture, we provide a brief introduction to polaron-type models, where a particle is linearly coupled to a non-relativistic quantum field. Our focus is on physically relevant models such as the Fröhlich polaron and the Nelson model. In the first part, we cover basic definitions, including the relevant Hilbert spaces, the form of the Hamiltonian, and some of its mathematical properties. In the second part, we investigate the role of translational symmetry in such particle-field models and explore the related question of self-localization. Self-localization refers to the possibility of the existence of a degenerate ground state despite the underlying symmetry of the Hamiltonian. Physically, this idea was proposed by Landau in 1933, suggesting that an electron could potentially get self-trapped through the interaction with the vibrations of a crystal lattice. We argue against the existence of ground states for translation-invariant particle-field models and, thus, against the phenomenon of self-localization in such a strict sense. If time permits, we provide a sketch of a rigorous proof specific to the Fröhlich polaron, based on the recent article (Lampart, Mitrouskas, Mysliwy, MPAG 26, 17, 2023).
January 10th, 2024
Ian Jauslin (Rutgers University)
Typicality in Statistical Mechanics and the arrow of time
In this lecture, I will give a brief overview of the foundations of statistical mechanics, with a focus on typicality in classical settings, and hint at generalizations to the quantum setting. I will also discuss open problems in equilibrium statistical mechanics.
March 13th, 2024
Lea Boßmann (JGU Mainz)
A brief introduction to the interacting Bose gas
Since the first experimental realization of a Bose-Einstein condensate in 1995, the physical and mathematical analysis of the Bose gas has become a very active field of research. In this lecture, I will give a short introduction to the topic from a mathematical perspective. I will focus on spectral properties of interacting Bose gases, in particular on their ground state energy and low-energy excitation spectrum.
April 24th, 2024
Jonas Lampart (Université de Bourgogne)
Contact interactions and generalised boundary conditions
Hamiltonians for a particle interacting with a point-like obstacle can be constructed as self-adjoint extensions of the Laplacian restricted to functions vanishing near the obstacle. These are characterised by a generalised boundary condition. They can be embedded into a larger family of Hamiltonians that also allow the particle to be absorbed or emitted at the obstacle.
After explaining these elementary constructions in detail, I will outline some generalisations to non-relativistic models in quantum field theory.
May 29th, 2024
Christian Brennecke (Universität Bonn)
An Introduction to the SK Model, Classical and Quantum
In this talk, I give an introduction to the classical SK model and some of its variants, including a quantum version with a transverse field. I discuss basics on replica symmetry breaking, the Parisi formula and some open questions related to the RS-RSB transition. The goal of the talk is to prepare the audience for a detailed discussion of a quantum version of the Parisi formula.
