The Geometry of Probabilistic Theories Convexity Representations and Information Processing

In the general probabilistic theory (GPT) approach to formulating candidate physical theories, systems are represented by compact convex sets of states, and measurement outcomes by affine functionals on the state space, with evaluation yielding probabilities. We will survey the theory of convexity, ordered linear spaces and their duality, and positive linear maps needed to describe such systems. We will focus on state spaces with interesting structural properties, such as: weak and strong self-duality, abstractions of the spectral decomposition of density matrices, conditions on the lattice of faces of the state space, and various symmetry properties. We'll discuss how to combine such systems into composite systems while preserving such properties. Implications for information processing (cloning, broadcasting, teleportation, cryptography, information disturbance tradeoffs, -nonlocal- correlations, computation and more), thermodynamics, computation, and the strength of Bell-type correlations will be explored. Characterizations of the quantum formalism of density matrices and POVMs in terms of structural and/or information-processing properties within the broader context of GPTs, will be surveyed. Representation theory of groups of automorphisms of the relevant convex sets is crucial to many of these topics, making GPTs a source of natural and concrete open problems deeply related to an emerging research frontier in pure mathematics at the nexus of convexity, algebraic geometry, and representation theory. Applications in optimization, partial differential equations, and statistics may also be touched on. Physik, Master of Science Modul: Ausgewählte Themen moderner Physik Universität Hannover SoSe 2016 Fakultät für Mathematik und Physik Dozent