Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory(Mathematical Surveys and Monographs)

The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach 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, especially the part that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.

Table of Contents:
Alice and Bob: Mathematical Aspects of Quantum Information: Notation and basic concepts Elementary convex analysis The mathematics of quantum information theory Quantum mechanics for mathematicians Banach and His spaces: Asymptotic Geometric Analysis Miscellany: More convexity Metric entropy and concentration of measure in classical spaces Gaussian processes and random matrices Some tools from asymptotic geometric analysis The Meeting: AGA and QIT: Entanglement of pure states in high dimensions Geometry of the set of mixed states Random quantum states Bell inequalities and the Grothendieck-Tsirelson inequality POVMs and the distillability problem Gaussian measures and Gaussian variables Classical groups and manifolds Extreme maps between Lorentz cones and the $S$-lemma Polarity and the Santalo point via duality of cones Hints to exercises Bibliography Notation Index.

About the Author :
Guillaume Aubrun, Universite Claude Bernard Lyon 1, Villeurbanne, France. Stanislaw J. Szarek, Case Western Reserve University, Cleveland, OH.

Review :
[This book] will be an invaluable reference to scientists working on the more mathematical aspects of quantum information theory." — Mary Beth Ruskai, Zentralblatt MATH "A wide variety of audiences would be interested in this book: Parts II or III would be suitable for a graduate course on QIT from the perspective of functional analysis, convex geometry, or random matrix theory, or on the applications of AGA. With a mix of classical and recent results, as well as the concise treatment of the subject areas, the book could be used as a reference book for researchers working in this area. Furthermore, the large number of exercises, with an appendix of hints, would make it suitable for an independent study." — Sarah Plosker, Mathematical Reviews