**Mathematical Surveys and Monographs**

Volume: 223;
2017;
414 pp;
Hardcover

MSC: Primary 46; 52; 81; 60;

Print ISBN: 978-1-4704-3468-7

Product Code: SURV/223

List Price: $116.00

AMS Member Price: $92.80

MAA Member Price: $104.40

**Electronic ISBN: 978-1-4704-4172-2
Product Code: SURV/223.E**

List Price: $116.00

AMS Member Price: $92.80

MAA Member Price: $104.40

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#### Supplemental Materials

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

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*Guillaume Aubrun; Stanisław J. Szarek*

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, 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. 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.

#### Readership

Graduate students and researchers interested in mathematical aspects of quantum information theory and quantum computing.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

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

Table of Contents pages: 1 2

- Chapter 10. Random quantum states 263286
- Chapter 11. Bell inequalities and the Grothendieck–Tsirelson inequality 275298
- Chapter 12. POVMs and the distillability problem 299322
- Appendix A. Gaussian measures and Gaussian variables 307330
- Appendix B. Classical groups and manifolds 311334
- Appendix C. Extreme maps between Lorentz cones and the S-lemma 321344
- Appendix D. Polarity and the Santaló point via duality of cones 325348
- Appendix E. Hints to exercises 329352
- Appendix F. Notation 375398
- Bibliography 381404
- Index 409432

- Back Cover Back Cover1442

Table of Contents pages: 1 2