Hardcover ISBN:  9781470434687 
Product Code:  SURV/223 
List Price:  $116.00 
MAA Member Price:  $104.40 
AMS Member Price:  $92.80 
Electronic ISBN:  9781470441722 
Product Code:  SURV/223.E 
List Price:  $116.00 
MAA Member Price:  $104.40 
AMS Member Price:  $92.80 

Book DetailsMathematical Surveys and MonographsVolume: 223; 2017; 414 ppMSC: Primary 46; 52; 81; 60;
The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twentyfirst 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, highdimensional 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 userfriendly 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.ReadershipGraduate students and researchers interested in mathematical aspects of quantum information theory and quantum computing.

Table of 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 GrothendieckTsirelson 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


Additional Material

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


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The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twentyfirst 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, highdimensional 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 userfriendly 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 GrothendieckTsirelson 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