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Full Description
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.
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.
