Quantum Symmetries: Approximation Properties, Operator Algebras and Applications to Quantum Information
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This research project aims to address several fundamental problems arising in two areas of mathematical analysis that have have their origins in quantum physics: operator algebra theory and quantum information theory. The common mathematical structures that thread the various aspects of this project together are called quantum symmetries. Quantum symmetry can be thought of as an enriched notion of symmetry, adapted to model the complex structures arising in nature. In practical contexts, mathematicians and scientists are interested in studying properties of a given system (e.g., a mechanical system, a molecule, or a population of organisms). Understanding the symmetries of such a system - transformations of the system that preserve the relevant structure - provides a wealth of insight into its properties. In many contexts, the symmetries that one encounters are described by mathematical structures called groups. However, for certain complex systems (particularly those connected to quantum mechanical phenomena, or operator algebra theory), groups are insufficient to describe the relevant symmetries that arise. This results in the need for a more general mathematical notion of symmetry, and led to the concept of a quantum group. Quantum groups were formally introduced in the 1980''s as a tool to study certain types of non-classical symmetries arising in statistical physics. Since that time, quantum groups (and related notions of quantum symmetry) have proved to have remarkable applications in topology, quantum algebra, operator algebras, quantum probability, quantum information theory, and the classification of topological phases of matter. By exploiting the quantum symmetries appearing in various mathematical and physical systems, this project aims to provide new insights into challenging problems in both operator algebra theory and quantum information theory. This project is divided into three components. The first part concerns the structural theory of von Neumman algebras associated to a class of discrete quantum groups, called orthogonal free quantum groups. The principle aim here is to use recent exciting developments in free probability theory and geometric quantum group theory to solve a conjecture which states that these von Neumann algebras can never be isomorphic to free group factors. The second part of this project concerns the structure of a class of von Neumann algebras, called the q-deformed Araki-Woods algebras. The PI will use certain asymptotic distributional symmetries for these algebras to investigate finite dimensional approximation properties and algebraic indecomposability results. The third and final part of this project proposes completely new applications of quantum symmetries in quantum information theory (QIT). The PI will investigate a new method for constructing examples of quantum channels using representation categories of quantum groups. Applications of these new quantum channels to various relevant questions in QIT will be investigated.