An error was encountered while trying to add the item to the cart. Please try again.
Copy To Clipboard
Successfully Copied!
Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory

Guillaume Aubrun Université Claude Bernard Lyon 1, Villeurbanne, France
Stanisław J. Szarek Case Western Reserve University, Cleveland, OH and Sorbonne Université , Paris, France
Available Formats:
Hardcover ISBN: 978-1-4704-3468-7
Product Code: SURV/223
List Price: $116.00 MAA Member Price:$104.40
AMS Member Price: $92.80 Electronic ISBN: 978-1-4704-4172-2 Product Code: SURV/223.E List Price:$116.00
MAA Member Price: $104.40 AMS Member Price:$92.80
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $174.00 MAA Member Price:$156.60
AMS Member Price: $139.20 Click above image for expanded view Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory Guillaume Aubrun Université Claude Bernard Lyon 1, Villeurbanne, France Stanisław J. Szarek Case Western Reserve University, Cleveland, OH and Sorbonne Université , Paris, France Available Formats:  Hardcover ISBN: 978-1-4704-3468-7 Product Code: SURV/223  List Price:$116.00 MAA Member Price: $104.40 AMS Member Price:$92.80
 Electronic ISBN: 978-1-4704-4172-2 Product Code: SURV/223.E
 List Price: $116.00 MAA Member Price:$104.40 AMS Member Price: $92.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$174.00 MAA Member Price: $156.60 AMS Member Price:$139.20
• Book Details

Mathematical Surveys and Monographs
Volume: 2232017; 414 pp
MSC: Primary 46; 52; 81; 60;

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.

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

• 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

• Reviews

• [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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2232017; 414 pp
MSC: Primary 46; 52; 81; 60;

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.

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

• 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
• [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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
You may be interested in...
Please select which format for which you are requesting permissions.