On quantum randomness and quantum resources

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Physics, 2018Cataloged from PDF version of thesis.Includes bibliographical references.This thesis is consisted of two independent parts. The first part is on entanglement, quantum randomness, and complexity beyond scrambling. More explicitly, we study the Rényi entanglement entropies of quantum designs. The results lay the mathematical foundation for studying the hierarchy of complexities in between scrambling and Haar randomness by entanglement. The second part explores the general aspects of quantum resource theory. We introduce three theories that do not rely on the specific resource: the theory of resource destroying maps, the one-shot operational resource theory, and the resource theory of quantum channels.by Zi-Wen Liu.Ph. D.Ph.D. Massachusetts Institute of Technology, Department of Physic