Softcover ISBN:  9781470451363 
Product Code:  CBMS/132 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
eBook ISBN:  9781470452919 
Product Code:  CBMS/132.E 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
Softcover ISBN:  9781470451363 
eBook: ISBN:  9781470452919 
Product Code:  CBMS/132.B 
List Price:  $116.00 $87.00 
MAA Member Price:  $104.40 $78.30 
AMS Member Price:  $92.80 $69.60 
Softcover ISBN:  9781470451363 
Product Code:  CBMS/132 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
eBook ISBN:  9781470452919 
Product Code:  CBMS/132.E 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
Softcover ISBN:  9781470451363 
eBook ISBN:  9781470452919 
Product Code:  CBMS/132.B 
List Price:  $116.00 $87.00 
MAA Member Price:  $104.40 $78.30 
AMS Member Price:  $92.80 $69.60 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 132; 2019; 144 ppMSC: Primary 68; 14; 15; 81
Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. It begins with an introduction to the geometry of tensors and provides geometric expositions of the basics of quantum information theory, Strassen's laser method for matrix multiplication, and moment maps in algebraic geometry. It also details several exciting recent developments regarding tensors in general. In particular, it discusses and explains the following material previously only available in the original research papers: (1) Shitov's 2017 refutation of longstanding conjectures of Strassen on rank additivity and Common on symmetric rank; (2) The 2017 ChristandlVranaZuiddam quantum spectral points that bring together quantum information theory, the asymptotic geometry of tensors, matrix multiplication complexity, and moment polytopes in geometric invariant theory; (3) the use of representation theory in quantum information theory, including the solution of the quantum marginal problem; (4) the use of tensor network states in solid state physics, and (5) recent geometric paths towards upper bounds for the complexity of matrix multiplication.
Numerous open problems appropriate for graduate students and postdocs are included throughout.
ReadershipUndergraduate and graduate students and researchers interested in tensors, mathematical aspects of solid state physics, and quantum information theory.

Table of Contents

Basics

Motivation, first definitions and properties

Tensors via linear algebra

Rank and border rank

Tensor networks

The asymptotic geometry of tensors

Detour into probability and information theory

Strassen’s laser method and spectral theory

Quantum mechanics for quantum information theory

Quantum information theory and the asymptotic geometry of tensors

Moment maps and moment polytopes

Hints and answers to selected exercises


Additional Material

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Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. It begins with an introduction to the geometry of tensors and provides geometric expositions of the basics of quantum information theory, Strassen's laser method for matrix multiplication, and moment maps in algebraic geometry. It also details several exciting recent developments regarding tensors in general. In particular, it discusses and explains the following material previously only available in the original research papers: (1) Shitov's 2017 refutation of longstanding conjectures of Strassen on rank additivity and Common on symmetric rank; (2) The 2017 ChristandlVranaZuiddam quantum spectral points that bring together quantum information theory, the asymptotic geometry of tensors, matrix multiplication complexity, and moment polytopes in geometric invariant theory; (3) the use of representation theory in quantum information theory, including the solution of the quantum marginal problem; (4) the use of tensor network states in solid state physics, and (5) recent geometric paths towards upper bounds for the complexity of matrix multiplication.
Numerous open problems appropriate for graduate students and postdocs are included throughout.
Undergraduate and graduate students and researchers interested in tensors, mathematical aspects of solid state physics, and quantum information theory.

Basics

Motivation, first definitions and properties

Tensors via linear algebra

Rank and border rank

Tensor networks

The asymptotic geometry of tensors

Detour into probability and information theory

Strassen’s laser method and spectral theory

Quantum mechanics for quantum information theory

Quantum information theory and the asymptotic geometry of tensors

Moment maps and moment polytopes

Hints and answers to selected exercises