Quantum entropy and its use

Entropy is a central quantity in information theory, probability and physics. This spring school will focus on fundamental concepts and basic operational interpretations of entropy with a particular focus on applications to quantum mechanics. The goal is provide a thorough overview ranging from fundamental concepts to present-day research problems in the field. A first introductory part of the spring school is devoted to the concept of entropy in classical information theory. We will study data processing, channels and their capacities and Shannon’s channel coding theorem. Historically, entropy was introduced by Boltzmann in the context of classical statistical mechanics. The mathematical key to understanding the equivalence of statistical ensembles are the laws of large numbers and large deviation principles. We will study these concepts on the basis of two examples: the asymptotic equipartition property in information theory and Cramer’s large deviation principle with an application to statistical mechanics. The second, major part of the spring school focuses on basic concepts in quantum information theory. After a short review of fundamental notions of quantum mechanics, we will focus on information transmission over quantum channels. We discuss major results in quantum (Shannon) theory: the Holevo-Schumacher-Westmoreland theorem which concerns the classical capacity of a quantum channel, as well as the Lloyd-Shor-Devetak theorem for the quantum capacity. We also consider the remarkable phenomenon of (super)activation of capacities, a purely quantum effect. In the last part of the school, we will discuss the role of entropy in many-body physics, including, in particular, area laws for mutual information, and statements about the approximability by tensor network states. Format: The format of the school is as follows. Each participant gives a talk about a preselected topic and there will be plenty of room for discussions and questions concerning background and open questions. Talks will be prepared in advance. We will be available to assist with the preparation. Prerequisites: A keen interest in the emergence of quantum effects in operational settings. Some familiarity with basic notions of quantum mechanics is desirable. Registration: by email to [email protected]. Spots are limited and will be assigned on a first-come-first-served basis.