# Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory

### About this Title

**Guillaume Aubrun**, *Université Claude Bernard Lyon 1, Villeurbanne, France* and **Stanisław J. Szarek**, *Case Western Reserve University, Cleveland, OH*

Publication: Mathematical Surveys and Monographs

Publication Year:
2017; Volume 223

ISBNs: 978-1-4704-3468-7 (print); 978-1-4704-4172-2 (online)

DOI: https://doi.org/10.1090/surv/223

MathSciNet review: MR3699754

MSC: Primary 46-02; Secondary 46B20, 52A07, 81-02, 81P45, 81R50, 94-02, 94A17

### Table of Contents

**Front/Back Matter**

**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 Santaló point via duality of cones
- Hints to exercises
- Notation