E-Book Overview
This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.
E-Book Content
The Theory of Quantum Information John Watrous Institute for Quantum Computing University of Waterloo
©2018 John Watrous To be published by Cambridge University Press. Please note that this is a draft, pre-publication copy only. The final, published version of this book will be available for purchase through Cambridge University Press and other standard distribution channels. This draft copy is made available for personal use only and must not be sold or redistributed.
Contents
Preface
page vii
1
Mathematical preliminaries 1.1 Linear algebra 1.1.1 Complex Euclidean spaces 1.1.2 Linear operators 1.1.3 Operator decompositions and norms 1.2 Analysis, convexity, and probability theory 1.2.1 Analysis and convexity 1.2.2 Probability theory 1.2.3 Semidefinite programming 1.3 Suggested references
2
Basic notions of quantum information 2.1 Registers and states 2.1.1 Registers and classical state sets 2.1.2 Quantum states of registers 2.1.3 Reductions and purifications of quantum states 2.2 Quantum channels 2.2.1 Definitions and basic notions concerning channels 2.2.2 Representations and characterizations of channels 2.2.3 Examples of channels and other mappings 2.2.4 Extremal channels 2.3 Measurements 2.3.1 Two equivalent definitions of measurements 2.3.2 Basic notions concerning measurements 2.3.3 Extremal measurements and ensembles 2.4 Exercises 2.5 Bibliographic remarks
1 1 1 8 24 35 35 47 53 57 58 58 58 61 67 72 72 77 91 96 100 100 105 113 120 122
iv
3
Contents
Similarity and distance among states and channels 3.1 Quantum state discrimination 3.1.1 Discriminating between pairs of quantum states 3.1.2 Discriminating quantum states of an ensemble 3.2 The fidelity function 3.2.1 Elementary properties of the fidelity function 3.2.2 Characterizations of the fidelity function 3.2.3 Further properties of the fidelity function 3.3 Channel distances and discrimination 3.3.1 Channel discrimination 3.3.2 The completely bounded trace norm 3.3.3 Distances between channels 3.3.4 Characterizations of the completely bounded trace norm 3.4 Exercises 3.5 Bibliographic remarks
185 197 198
4
Unital channels and majorization 4.1 Subclasses of unital channels 4.1.1 Mixed-unitary channels 4.1.2 Weyl-covariant channels 4.1.3 Schur channels 4.2 General properties of unital channels 4.2.1 Extreme points of the set of unital channels 4.2.2 Fixed-points, spectra, and norms of unital channels 4.3 Majorization 4.3.1 Majorization for real vectors 4.3.2 Majorization for Hermitian operators 4.4 Exercises 4.5 Bibliographic remarks
201 201 202 212 219 222 222 228 233 233 241 246 247
5
Quantum entropy and source coding 5.1 Classical entropy 5.1.1 Definitions of classical entropic function