Geometric and information-theoretic aspects of quantum thermodynamics

In this thesis, I investigate various aspects of one of the most fundamental questions in thermodynamics: what state transformations can quantum systems undergo while interacting with a thermal bath under specific constraints? These constraints may involve total energy conservation, memory effects, or finite-size considerations. Addressing this question leads to (i) a characterisation of the structure of the thermodynamic arrow of time, (ii) a framework bridging the gap between memoryless and arbitrarily non-Markovian thermodynamic processes, and (iii) a derivation of the famous fluctuation-dissipation relation within a quantum information framework. Finally, the last part of this thesis focuses on studying a ubiquitous phenomenon in science, so-called catalysis. It involves using an auxiliary system (a catalyst) to enable processes that would otherwise be impossible. Over the last two decades, this notion has spread to the field of quantum physics. However, this effect is typically described within a highly abstract framework. Despite its successes, this approach struggles to fully capture the behaviour of physically realisable systems, thereby limiting the applicability of quantum catalysis in practical scenarios. Strikingly, I will demonstrate this effect in a paradigmatic quantum optics setup, namely the Jaynes-Cummings model, where an atom interacts with an optical cavity. The atom plays the role of the catalyst and allows for the deterministic generation of non-classical light in the cavity, as evidenced by sub-Poissonian statistics or Wigner negativity.