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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
Alice and Bob Meet Banach
Softcover ISBN:  978-1-4704-7796-7
Product Code:  SURV/223.S
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Not yet published - Preorder Now!
Expected availability date: June 30, 2024
eBook ISBN:  978-1-4704-4172-2
Product Code:  SURV/223.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN:  978-1-4704-7796-7
eBook: ISBN:  978-1-4704-4172-2
Product Code:  SURV/223.S.B
List Price: $250.00 $187.50
MAA Member Price: $225.00 $168.75
AMS Member Price: $200.00 $150.00
Not yet published - Preorder Now!
Expected availability date: June 30, 2024
Alice and Bob Meet Banach
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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
Softcover ISBN:  978-1-4704-7796-7
Product Code:  SURV/223.S
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Not yet published - Preorder Now!
Expected availability date: June 30, 2024
eBook ISBN:  978-1-4704-4172-2
Product Code:  SURV/223.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN:  978-1-4704-7796-7
eBook ISBN:  978-1-4704-4172-2
Product Code:  SURV/223.S.B
List Price: $250.00 $187.50
MAA Member Price: $225.00 $168.75
AMS Member Price: $200.00 $150.00
Not yet published - Preorder Now!
Expected availability date: June 30, 2024
  • 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.

    Readership

    Graduate 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 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 publishers of book reviews
    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.

Readership

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 publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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