The Functional Analysis of Quantum Information Theory
Overview
- Authors:
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Ved Prakash Gupta
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School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India
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Prabha Mandayam
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Department of Physics, Indian Institute of Technology Madras, Chennai, India
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V.S. Sunder
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The Institute of Mathematical Sciences, Chennai, India
- Authored by leading researchers in the field
- First broad yet concise introduction to the subject matter
- Tutorial and self-contained presentation
Table of contents (4 chapters)
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- Ved Prakash Gupta, Prabha Mandayam, V. S. Sunder
Pages 1-37
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- Ved Prakash Gupta, Prabha Mandayam, V. S. Sunder
Pages 39-62
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- Ved Prakash Gupta, Prabha Mandayam, V. S. Sunder
Pages 63-92
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- Ved Prakash Gupta, Prabha Mandayam, V. S. Sunder
Pages 93-134
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Back Matter
Pages 135-139
About this book
This book provides readers with a concise introduction to current studies on operator-algebras and their generalizations, operator spaces and operator systems, with a special focus on their application in quantum information science. This basic framework for the mathematical formulation of quantum information can be traced back to the mathematical work of John von Neumann, one of the pioneers of operator algebras, which forms the underpinning of most current mathematical treatments of the quantum theory, besides being one of the most dynamic areas of twentieth century functional analysis. Today, von Neumann’s foresight finds expression in the rapidly growing field of quantum information theory. These notes gather the content of lectures given by a very distinguished group of mathematicians and quantum information theorists, held at the IMSc in Chennai some years ago, and great care has been taken to present the material as a primer on the subject matter. Starting from the basic definitions of operator spaces and operator systems, this text proceeds to discuss several important theorems including Stinespring’s dilation theorem for completely positive maps and Kirchberg’s theorem on tensor products of C*-algebras. It also takes a closer look at the abstract characterization of operator systems and, motivated by the requirements of different tensor products in quantum information theory, the theory of tensor products in operator systems is discussed in detail. On the quantum information side, the book offers a rigorous treatment of quantifying entanglement in bipartite quantum systems, and moves on to review four different areas in which ideas from the theory of operator systems and operator algebras play a natural role: the issue of zero-error communication over quantum channels, the strong subadditivity property of quantum entropy, the different norms on quantum states and the corresponding induced norms on quantum channels, and, lastly, the applications of matrix-valued random variables in the quantum information setting.
Reviews
“This volume is a collection of notes from a two-week workshop … . the contributions from the four workshop speakers are quite well written and the editors have achieved a high level of consistency in style and terminology. … This collection will be of interest to researchers in physics, mathematics, and theoretical computer science.” (Kevin J. Compton, Mathematical Reviews, January, 2016)
Authors and Affiliations
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School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India
Ved Prakash Gupta
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Department of Physics, Indian Institute of Technology Madras, Chennai, India
Prabha Mandayam
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The Institute of Mathematical Sciences, Chennai, India
V.S. Sunder
