Alice and Bob Meet Banach

This book builds a bridge between two scientific areas. The first area, Asymptotic Geometric Analysis (AGA), studies the geometry of Banach spaces through their subspaces or subsets of finite but large dimension. Geometric and probabilistic techniques, such as concentration of measure, play a fundamental role. The second area, Quantum Information Theory (QIT), provides the mathematical framework for manipulation of information in the quantum world, and for using quantum phenomena to transmit or store data. Both fields are given a detailed presentation, which includes a discussion of selected important results. One of the main goals of the book is to show how by combining these two areas, one can get deep recent results about the geometry of entanglement or superadditivity of quantum channel capacities. The book is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, or at least the part of QIT that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. The book aims at making the relevant concepts and facts accessible to everyone from a casual reader to an expert looking for a reference.