Galois Fields and Galois Rings Made Easy
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The existing literature on rings and fields is primarily mathematical. There are a great number of excellent books on the theory of rings and fields written by and for mathematicians, but these can be difficult for physicists and chemists to access.
This book offers an introduction to rings and fields with numerous examples. It contains an application to the construction of mutually unbiased bases of pivotal importance in quantum information. It is intended for graduate and undergraduate students and researchers in physics, mathematical physics and quantum chemistry (especially in the domains of advanced quantum mechanics, quantum optics, quantum information theory, classical and quantum computing, and computer engineering).
Although the book is not written for mathematicians, given the large number of examples discussed, it may also be of interest to undergraduate students in mathematics.
- Contains numerous examples that accompany the text
- Includes an important chapter on mutually unbiased bases
- Helps physicists and theoretical chemists understand this area of mathematics
Maurice Kibler
Maurice Kibler is Professor Emeritus at Claude Bernard University Lyon 1 in France. His research concerns the role of symmetries in the elaboration of models in various domains of physics (sub-atomic, atomic, molecular and condensed matter physics). He is also interested in quantum information.
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Galois Fields and Galois Rings Made Easy - Maurice Kibler
Galois Fields and Galois Rings Made Easy
Maurice R. Kibler
Table of Contents
Cover
Title page
Dedication
Copyright
Acknowledgments
Preface
List of Mathematical Symbols
Sets
Numbers
Matrices
Groups
Rings
Fields
1: The Structures of Ring and Field
Abstract
1.1 Rings
1.2 Fields
2: Galois Fields
Abstract
2.1 Generalities
2.2 Extension of a field: a typical example
2.3 Extension of a field: the general case
2.4 Sub-field of a Galois field
2.5 Factorizations
2.6 The application trace for a Galois field
2.7 Bases of a Galois field
2.8 Characters of a Galois field
2.9 Gaussian sums over Galois fields
3: Galois Rings
Abstract
3.1 Generalities
3.2 Construction of a Galois ring
3.3 Examples and counter-examples of Galois rings
3.4 The application trace for a Galois ring
3.5 Characters of a Galois ring
3.6 Gaussian sums over Galois rings
4: Mutually Unbiased Bases
Abstract
4.1 Generalities
4.2 Quantum angular momentum bases
4.3 SU(2) approach to mutually unbiased bases
4.4 Galois field approach to mutually unbiased bases
4.5 Galois ring approach to mutually unbiased bases
5: Appendix on Number Theory and Group Theory
Abstract
5.1 Elements of number theory
5.2 Elements of group theory
Bibliography
Index
Dedication
To my granddaughter Éloïse Kibler Blau
Copyright
First published 2017 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Press Ltd
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Elsevier Ltd
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Notices
Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.
For information on all our publications visit our website at http://store.elsevier.com/
© ISTE Press Ltd 2017
The rights of Maurice R. Kibler to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
A catalog record for this book is available from the Library of Congress
ISBN 978-1-78548-235-9
Printed and bound in the UK and US
Acknowledgments
I am indebted to Natig Atakishiyev, Mohammed Daoud, Serge Perrine, Michel Planat, Metod Saniga and Bernardo Wolf, as well as, last (but not the least), my student Olivier Albouy for numerous discussions, to Apostol Vourdas for discussions and e-mail correspondence on Galois quantum mechanics, to Bruce Berndt, Ron Evans and Philippe Langevin for e-mail correspondence on quadratic Gauss sums, to Markus Grassl, Arthur Pittenger and Stefan Weigert for e-mail correspondence on MUBs, and to Philippe Caldero for a reading of the manuscript and providing valuable comments.
Finally, I am grateful to my wife Gloria for her patience and continual encouragement in the course of writing this book.
Preface
Maurice Kibler June 2017
This book constitutes an elementary introduction to rings and fields, especially Galois rings and Galois fields, with regard to their application to the theory of quantum information.
Since the 1930s, the theory of groups has been widely used in many domains of physical sciences (elementary particle and nuclear physics, atomic and molecular physics, condensed matter physics, theoretical and quantum chemistry). In contrast, the theory of rings and fields, which comes immediately after group theory in the hierarchy of abstract algebra, is less well known to physicists and chemists. Of course, fields with an infinite number of elements like the field of real numbers, the field of complex numbers and, to some extent, the field of quaternions are all well known in the physical sciences. Similarly, infinite rings, such as the ring of integers and the ring of square matrices, are of common usage in physics and chemistry. However, finite rings and finite fields (largely used in pure mathematics and in the classical theory of information) are relatively unknown to physicists and chemists - despite their potential utility for the quantum theory of information, having been recognized in the 1990s.
The existing literature on rings and fields is primarily mathematical. There are a great deal of excellent books on the theory of rings and fields written by and for mathematicians, but these can be difficult for physicists and chemists to access. The present book offers an introduction to rings and fields for students and researchers in physics and chemistry, with an emphasis on their application to the construction of mutually unbiased bases of pivotal importance in quantum information. This book is intended for graduate and undergraduate students and researchers in physics, mathematical physics and quantum chemistry (especially in the domains of advanced quantum mechanics, quantum optics, quantum information theory, classical and quantum computing, and computer engineering). Although the book is not written for mathematicians, given the large number of examples discussed, it may be of interest to undergraduate students in mathematics.
The book is organized as follows. Chapter 1 is devoted to a general discussion of the algebraic structures of rings and fields. Chapters 2 and 3 deal with Galois fields (i.e. finite fields) and Galois rings (i.e. special finite rings) respectively. Chapter 4 is concerned with the construction of mutually unbiased bases in Hilbert spaces of finite dimension; for Hilbert spaces of dimension pm, with p a prime number and m a positive integer, Galois rings are used for p even and Galois fields for p odd. Finally, for the reader unfamiliar with number theory and group theory, an appendix (Chapter 5) lists some basic results that are necessary for the understanding of the first four chapters. Finally, a list of references (Bibliography) closes the book; this list includes some relevant web links. In sum, the book is divided into two parts: a mathematical part (Chapters 1, 2, 3 and 5) and a physical part (Chapter 4). Some further specification of the two parts is in order.
The presentation and the pedagogy of the mathematical part of this book differ in many respects from those encountered in books dealing with pure mathematics. This part may be considered as a collection of definitions and results (propositions and properties) useful to the practitioner. The emphasis is placed on examples (with numerous repetitions), that sometimes come before the results, rather than on proofs of the results. Nevertheless, some proofs and some basic elements of proof are given. More involved proofs can be found in the books listed in the sections Mathematical literature: rings and fields and Mathematical literature: Theory of numbers of the bibliography at the end of the book. In this respect, the seminal textbooks by B.C. Berndt, R.J. Evans and K.S. Williams, M. Demazure, L.K. Hua, R. Lidl and H. Niederreiter, and Z.-X. Wan were of invaluable help in writing the present book. The reader is encouraged to consult these books not only for their mathematical content but also for their impressive lists of references and historical notes.
For the physical part of this book, a basic knowledge of quantum mechanics is required. The concept of a set of mutually unbiased bases (MUBs for short) has been the subject of great activity since the end of the 1990s (a nonexhaustive list of references is given in the section Theoretical physics literature: MUBs). The concept of MUBs is introduced in for p , with m is shown as an application of the principles discussed in Chapters 2 and 3.
To conclude, let us mention that the calculations arising in the numerous examples can be achieved very easily by using a symbolic and numeric programming language such as Maple.
List of Mathematical Symbols
Sets
means for any
means there exists
→ depending on the context, the vertical bar | may mean such that (for example, α | f (α) = 0 means α is such that f (α) = 0)or divides (like in b | a which means b divides a(which denotes an inner product in a pre-Hilbert space)
) means that the set B is a subset of Acan be: B is a sub-group of A or B is a sub-ring of A or B is a sub-field of A
denotes the union of the sets A and B
, then A \ B denotes the set A from which the elements of the set B are missing
→ Card(S) denotes the cardinal of the set S
Numbers
in the French literature)
in the French literature)
comprises the negative integers, 0 and the positive integers
→ d
is the set of rational numbers
is the set of real numbers
is the set of complex numbers
is the set of complex numbers without 0
→ n
→ e stands for the basis of Napierian logarithms
stands for the pure imaginary number
→ |z
→ i, j, k are the basic quaternions
→ p is an even or odd prime (the letter p is used for denoting prime numbers)
→ pm
means that the integer a divides the integer b
means that the integer a does not divide the integer b
; we say that a is congruent to b modulo n (in many places where the context modulo mod n is simply noted as a = b)
→ gcd(a, b) stands for the greatest common divisor of the integers a and b
→ (a, b) = 1 means that the integers a and b are co-prime (their greatest common divisor is 1)
→ δ is the Kronecker symbol (for m, δ(n, m) = 0 if n ≠ m and δ(n, m) = 1 if n = m)
→ φ denotes the Euler function
→ μ denotes the Möbius function
is the Legendre symbol (a integer and p prime)
→ Cpk is the Newton binomial coefficient
→ G(d) denotes the usual Gauss sum
Matrices
→ the sign ⊗ indicates the tensor product of vectors (in the framework of vector spaces)
→ det (A) denotes the determinant of the matrix A
→ tr (A) denotes the trace of the matrix A (tr is reserved for matrices and Tr for rings and fields)
→ A† stands for the Hermitian conjugate (i.e. transpose + complex conjugate) of the matrix A
→ [X, Y] or [X, Y]− denotes the commutator of the matrices (or operators) X and Y
→ [X, Y]+ denotes the anticommutator of the matrices (or operators) X and Y
Groups
→ |G| denotes the cardinal (or order) of the group G
→ ker (f) denotes the kernel of the group homomorphism f
→ G/H stands for the quotient group G by its normal sub-group H
means that the groups G and G′ are isomorphic (the symbol ≃ is reserved to groups)
is the direct product of the groups G and G′
→ V is the Klein four-group
→ Cd denotes the cyclic group of order d
→ 〈a〉 stands for the cyclic group generated by the element a
→ Sn is the symmetric group on n objects
→ An is the alternating group on n objects
→ GL (n) is the general linear group, in n
→ SL (n) is the special linear group, in n
→ O (n, R) is the orthogonal group, in n
→ SO (n, R) is the special orthogonal group, in n
→ U (n) is the unitary group, in n
→ SU (n) is the special unitary group, in n
Rings
→ the signs + and × (sometimes ⊕ and ⊗) denote the addition and multiplication laws of a ring, respectively
→ (R, +, ×) or simply R is used for denoting an arbitrary ring
denotes the direct product of the rings R and R′ (same notation as for groups)
→ charact (R) stands for the characteristic of the ring R
is the ring of integers
→ d
denotes the Galois ring of characteristic ps with psm elements (p prime and s)
→ R[ξ] denotes the ring of polynomials in the indeterminate ξ with coefficients in the ring R
denotes the ring of polynomials in the indeterminate ξ
→ 〈a〉 denotes a principal ideal spanned by the element a of a finite ring
→ Tr (a) denotes the trace of the element a of a ring
→ ϕ denotes the generalized Frobenius automorphism for a ring
→ χb , +) (χ0 is the trivial additive character)
→ ψk , ×) (ψ0 is the trivial multiplicative character)
→ Gm (ψk, χb) denotes the Gaussian sum for a Galois ring
Fields
is used for denoting an arbitrary field
is the field of rational numbers
is the field of real numbers
is the field of complex numbers
is the field of quaternions
denotes the Galois field of characteristic p with pm elements (p )
)
is the field of integers (or field of residues) modulo p (p prime)
)
denotes the ring of polynomials in the indeterminate ξ
denotes the ring of polynomials in the indeterminate ξ
→ Tr (x) denotes the trace of the element x of a field
→ σ denotes the Frobenius automorphism for a field
→ χy