**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

[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

#### 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

- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- List of Tables xiii14
- List of Figures xv16
- Preface xix20
- Part 1 . Alice and Bob Mathematical Aspects of Quantum Information Theory 124
- Chapter 0. Notation and basic concepts 326
- 0.1. Asymptotic and nonasymptotic notation 326
- 0.2. Euclidean and Hilbert spaces 326
- 0.3. Bra-ket notation 427
- 0.4. Tensor products 629
- 0.5. Complexification 629
- 0.6. Matrices vs. operators 730
- 0.7. Block matrices vs. operators on bipartite spaces 831
- 0.8. Operators vs. tensors 831
- 0.9. Operators vs. superoperators 831
- 0.10. States, classical and quantum 831

- Chapter 1. Elementary convex analysis 1134
- Chapter 2. The mathematics of quantum information theory 3154
- 2.1. On the geometry of the set of quantum states 3154
- 2.2. States on multipartite Hilbert spaces 3558
- 2.2.1. Partial trace 3558
- 2.2.2. Schmidt decomposition 3659
- 2.2.3. A fundamental dichotomy: Separability vs. entanglement 3760
- 2.2.4. Some examples of bipartite states 3962
- 2.2.5. Entanglement hierarchies 4164
- 2.2.6. Partial transposition 4164
- 2.2.7. PPT states 4366
- 2.2.8. Local unitaries and symmetries of Sep 4669

- 2.3. Superoperators and quantum channels 4770
- 2.4. Cones of QIT 5578
- Notes and Remarks 6386

- Chapter 3. Quantum mechanics for mathematicians 6790
- 3.1. Simple-minded quantum mechanics 6790
- 3.2. Finite vs. infinite dimension, projective spaces, and matrices 6891
- 3.3. Composite systems and quantum marginals: Mixed states 6891
- 3.4. The partial trace: Purification of mixed states 7093
- 3.5. Unitary evolution and quantum operations: The completely positive maps 7194
- 3.6. Other measurement schemes 7396
- 3.7. Local operations 7497
- 3.8. Spooky action at a distance 7598
- Notes and Remarks 7598

- Part 2 . Banach and His Spaces Asymptotic Geometric Analysis Miscellany 77100
- Chapter 4. More convexity 79102
- Chapter 5. Metric entropy and concentration of measure in classical spaces 107130
- 5.1. Nets and packings 107130
- 5.2. Concentration of measure 117140
- 5.2.1. A prime example: concentration on the sphere 119142
- 5.2.2. Gaussian concentration 121144
- 5.2.3. Concentration tricks and treats 124147
- 5.2.4. Geometric and analytic methods. Classical examples 129152
- 5.2.5. Some discrete settings 136159
- 5.2.6. Deviation inequalities for sums of independent random variables 139162

- Notes and Remarks 142165

- Chapter 6. Gaussian processes and random matrices 149172
- Chapter 7. Some tools from asymptotic geometric analysis 181204
- 7.1. ell-position, K-convexity and the MM*-estimate 181204
- 7.2. Sections of convex bodies 186209
- 7.2.1. Dvoretzky’s theorem for Lipschitz functions 186209
- 7.2.2. The Dvoretzky dimension 189212
- 7.2.3. The Figiel–Lindenstrauss–Milman inequality 193216
- 7.2.4. The Dvoretzky dimension of standard spaces 195218
- 7.2.5. Dvoretzky’s theorem for general convex bodies 200223
- 7.2.6. Related results 201224
- 7.2.7. Constructivity 205228

- Notes and Remarks 207230

- Part 3 . The Meeting: AGA and QIT 211234
- Chapter 8. Entanglement of pure states in high dimensions 213236
- 8.1. Entangled subspaces: Qualitative approach 213236
- 8.2. Entropies of entanglement and additivity questions 215238
- 8.3. Concentration of E-p for p > 1 and applications 218241
- 8.4. Concentration of von Neumann entropy and applications 222245
- 8.5. Entangled pure states in multipartite systems 229252
- Notes and Remarks 232255

- Chapter 9. Geometry of the set of mixed states 235258
- Chapter 10. Random quantum states 263286

Table of Contents pages: 1 2