Quantum Science and Technology

Apostolos Vourdas

Finite and Profinite Quantum Systems

Quantum Science and Technology

Series editors

Nicolas Gisin, Geneva, SwitzerlandRaymond Laflamme, Waterloo, CanadaGaby Lenhart, Sophia Antipolis, FranceDaniel Lidar, Los Angeles, USAGerard J. Milburn, St Lucia, AustraliaArno Rauschenbeutel, Vienna, AustriaRenato Renner, Zürich, SwitzerlandMaximilian Schlosshauer, Portland, USAYaakov S. Weinstein, Princeton, USAH.M. Wiseman, Brisbane, Australia

Aims and Scope

The book series Quantum Science and Technology is dedicated to one of today’smost active and rapidly expanding fields of research and development. In particular,the series will be a showcase for the growing number of experimental implemen-tations and practical applications of quantum systems. These will include, but arenot restricted to: quantum information processing, quantum computing, andquantum simulation; quantum communication and quantum cryptography; entan-glement and other quantum resources; quantum interfaces and hybrid quantumsystems; quantum memories and quantum repeaters; measurement-based quantumcontrol and quantum feedback; quantum nanomechanics, quantum optomechanicsand quantum transducers; quantum sensing and quantum metrology; as well asquantum effects in biology. Last but not least, the series will include books on thetheoretical and mathematical questions relevant to designing and understandingthese systems and devices, as well as foundational issues concerning the quantumphenomena themselves. Written and edited by leading experts, the treatments willbe designed for graduate students and other researchers already working in, orintending to enter the field of quantum science and technology.

More information about this series at http://www.springer.com/series/10039

Apostolos Vourdas

Finite and Profinite QuantumSystems

123

Apostolos VourdasDepartment of Computer ScienceUniversity of BradfordBradfordUK

ISSN 2364-9054 ISSN 2364-9062 (electronic)Quantum Science and TechnologyISBN 978-3-319-59494-1 ISBN 978-3-319-59495-8 (eBook)DOI 10.1007/978-3-319-59495-8

Library of Congress Control Number: 2017944304

© Springer International Publishing AG 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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Preface

Quantum mechanics is usually studied for systems where the position andmomentum take all real values. Weyl and later Schwinger studied quantum systemswhere the position and momentum take a finite number of values. The emergenceof the subject of quantum information created a lot of interest in such systems,which became known as qudits. This research is interdisciplinary and combinesideas from mathematics, physics, computer science, chemistry, and materials sci-ence. Within this general area, the so-called problem of mutually unbiased bases ledto the study of quantum systems with variables in Galois fields. The monographdiscusses various aspects of these ‘finite quantum systems.’

It also considers rigorously the limit where the dimension of the system becomesvery large using ideas from the timely area of profinite groups. These ‘profinitequantum systems’ have links to a different area of research, known as p-adicphysics. The latter studies quantum systems with variables which are p-adicnumbers and combines algebraic number theory with quantum physics. It hasapplications in condensed matter, particle physics, string theory, etc. The mono-graph approaches this area from a different angle, using inverse and direct limitsand profinite groups.

I have a strong interest in all these areas, and I have written three review articleson finite quantum systems (in 2004), on systems with variables in Galois fields (in2007), and on systems with variables which are p-adic numbers (in 2013). They arethe background for this monograph, but the material is completely rewritten, somegaps have been filled, and other topics such as finite geometries, mutually unbiasedbases, weak mutually unbiased bases, and quantum logic have been added.

Overall, the aim of this monograph is to present the material by adding a novelflavor to it (as discussed in section 1.4 below). The presentation is concise butinformative, and the general theory is complemented with examples. The level ofrigor is appropriate for a mathematical physics monograph. The proofs describe theimportant steps, so that the reader can easily fill the gaps.

The monograph is suitable for Ph.D. students and other researchers in quantumphysics, quantum optics, quantum information, p-adic physics, mathematical

v

physics, applied mathematics, and computer science. The reader is expected to havesome knowledge of certain mathematical areas, as follows:

• In Chap. 5, about finite geometries.• In Chap. 6, about lattice theory.• In Chaps. 8 and 9 about Galois theory.• In Chaps. 10, 11, and 12 about p-adic numbers, profinite groups, and inverse

and direct limits.

There is a brief introduction to these concepts in the monograph (with referencesto the literature), but the aim is to establish the notation and explain how to dopractical calculations. So some prior knowledge of these topics will be helpful.

I would like to thank some of my ex-Ph.D. students Dr. C. Lei, Dr. S. Zhang,Dr. L. Wang, Dr. H. Al Hadhrami, Dr. M. Shalaby, Dr. S. Oladejo,Dr. P. Evangelides, and Dr. T. Olupitan, who worked with me on related problems.I would also like to thank Tom Spicer and Cindy Zitter from Springer, and thereferee, for their encouragement and support.

Bradford, UK Apostolos VourdasApril 2017

vi Preface

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Finite Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Profinite Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Brief Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Aim of the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Partial Orders and Pontryagin Duality . . . . . . . . . . . . . . . . . . . . . . . 72.1 Partial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The Directed-Complete Partial Order of Supernatural

(Steinitz) Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Pontryagin Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 The Ring ZðdÞ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1 The Ring ZðdÞ and its Characters . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Quadratic Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Totient Functions and the Dedekind psi Function . . . . 123.1.3 Invertible and Non-invertible Elements in ZðdÞ . . . . . 13

3.2 Factorization of the Ring ZðdÞ Based on the ChineseRemainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 The Symplectic Group Sp½2;ZðdÞ� . . . . . . . . . . . . . . . . . . . . . . 163.4 Factorization of the Symplectic Group Sp½2;ZðdÞ�

Based on the Chinese Remainder Theorem . . . . . . . . . . . . . . . 17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Quantum Systems with Variables in ZðdÞ. . . . . . . . . . . . . . . . . . . . . 234.1 Fourier Transforms in R½ZðdÞ� . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 The Heisenberg-Weyl Group HW ½ZðdÞ�. . . . . . . . . . . . . . . . . . 26

vii

4.4 Coherence in Finite Quantum Systems. . . . . . . . . . . . . . . . . . . 294.4.1 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4.2 Coherent Density Matrices . . . . . . . . . . . . . . . . . . . . . 304.4.3 Coherent Projectors of Rank n . . . . . . . . . . . . . . . . . . 31

4.5 Symplectic Transformations and the Sp½2;ZðdÞ� Group . . . . . . 334.6 The HWSp½ZðdÞ� Group of Displacements and Symplectic

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.7 Parity Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.8 Wigner and Weyl Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 414.9 Factorization of R½ZðdÞ� Based on the Chinese Remainder

Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.10 Analytic Representation of Finite Quantum Systems . . . . . . . . 49References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Finite Geometries and Mutually Unbiased Bases . . . . . . . . . . . . . . . 575.1 The ZðdÞ � ZðdÞ as a Non-near-linear Finite Geometry . . . . . 58

5.1.1 Symplectic Transformations in the Finite GeometryZðdÞ � ZðdÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1.2 Factorization of the Finite Geometry ZðdÞ � ZðdÞBased on the Chinese Remainder Theorem. . . . . . . . . 61

5.2 Mutually Unbiased Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2.1 Mutually Unbiased Bases in H½ZðpÞ� . . . . . . . . . . . . . 65

5.3 Weak Mutually Unbiased Bases and Duality with FiniteGeometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.4 Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Quantum Logic of Finite Quantum Systems . . . . . . . . . . . . . . . . . . . 776.1 Classical Logic: Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . 78

6.1.1 Boolean Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.1.2 Classical Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 The Modular Orthocomplemented Lattice LðdÞ . . . . . . . . . . . . 836.2.1 Quantum Versus Classical Disjunction . . . . . . . . . . . . 836.2.2 Projectors to the Subspaces . . . . . . . . . . . . . . . . . . . . . 846.2.3 The Modularity Property of LðdÞ . . . . . . . . . . . . . . . . 856.2.4 Non-distributivity of the Lattice LðdÞ . . . . . . . . . . . . . 866.2.5 Transpose Intervals in the Lattice LðdÞ. . . . . . . . . . . . 87

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.1 Angle States and Angular Momentum States . . . . . . . . . . . . . . 93

7.1.1 The Schwinger Representation . . . . . . . . . . . . . . . . . . 957.1.2 Angle States and Angular Momentum States

in HB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.1.3 Area Preserving Diffeomorphisms on a Sphere . . . . . . 97

viii Contents

7.2 Interferometry in Multimode Systems . . . . . . . . . . . . . . . . . . . 1007.2.1 Fourier Interferometry and Applications

to Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2.2 Other Types of Interferometry. . . . . . . . . . . . . . . . . . . 102

7.3 Orbital Angular Momentum States . . . . . . . . . . . . . . . . . . . . . . 1037.4 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8 Galois Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.1 The Galois Field GFðpeÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.2 Subfields of GFðpeÞ and Galois Groups . . . . . . . . . . . . . . . . . . 1088.3 Trace and Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.4 General Bases in GFðpeÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9 Quantum Systems with Variables in GFðpeÞ. . . . . . . . . . . . . . . . . . . 1199.1 Fourier Transforms in R½GFðpeÞ� . . . . . . . . . . . . . . . . . . . . . . . 120

9.1.1 Change of the Basis in GFðpeÞ . . . . . . . . . . . . . . . . . . 1229.1.2 Comparison of the System R½GFðpeÞ�

with the System Rf½ZðpÞ�eg . . . . . . . . . . . . . . . . . . . . 1229.2 Frobenius Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.3 The Heisenberg-Weyl Group HW ½GFðpeÞ�. . . . . . . . . . . . . . . . 1279.4 Symplectic Transformations and the Sp½2;GFðpeÞ� Group . . . . 1319.5 Parity Operators and Wigner and Weyl Functions . . . . . . . . . . 1369.6 Hamiltonians of Galois Quantum Systems and Time

Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389.6.2 Galois Systems with Frobenius Symmetry . . . . . . . . . 139

9.7 Mutually Unbiased Bases in H½GFðpeÞ� and Dualityto GFðpeÞ � GFðpeÞ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409.7.1 The Finite Geometry GFðpeÞ � GFðpeÞ . . . . . . . . . . . 141

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

10 p-adic Numbers and Profinite Groups. . . . . . . . . . . . . . . . . . . . . . . . 14510.1 The Field Qp and the Ring Zp . . . . . . . . . . . . . . . . . . . . . . . . . 145

10.1.1 Absolute Values of p-adic Numbers . . . . . . . . . . . . . . 14610.1.2 Additive Characters. . . . . . . . . . . . . . . . . . . . . . . . . . . 147

10.2 Qp=Zp as the Pontryagin Dual Group of Zp . . . . . . . . . . . . . . 147

10.3 The Group bZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

10.4 Q=Z as the Pontryagin Dual Group of bZ . . . . . . . . . . . . . . . . . 14910.4.1 Additive Characters. . . . . . . . . . . . . . . . . . . . . . . . . . . 15010.4.2 The Directed-Complete Partial Order of Subgroups

of Q=Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15110.5 Inverse and Direct Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Contents ix

10.6 The Profinite Group Zp as Inverse Limit . . . . . . . . . . . . . . . . . 15310.6.1 Zp as a Compact and Totally Disconnected

Topological Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 15410.7 Qp=Zp as Direct Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15510.8 A Complete Chain of Pontryagin Dual Pairs of Groups . . . . . . 156

10.9 The Profinite Group bZp as Inverse Limit . . . . . . . . . . . . . . . . . 157

10.9.1 bZp as a Compact and Totally DisconnectedTopological Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

10.10 Q=Z as Direct Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.11 A Directed-Complete Partial Order of Pontryagin Dual Pairs

of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

11 A Quantum System with Positions in the Profinite Group Zp . . . . . 16111.1 Locally Constant Functions with Compact Support . . . . . . . . . 16111.2 Integrals of Complex Functions on Qp . . . . . . . . . . . . . . . . . . . 16211.3 Integrals of Complex Functions on Qp=Zp and Weil

Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16411.3.1 Weil Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16611.3.2 Delta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

11.4 The Quantum System R½Zp; ðQp=ZpÞ� . . . . . . . . . . . . . . . . . . . 16811.5 The Heisenberg-Weyl Group HW ½ðQp=ZpÞ;Zp; ðQp=ZpÞ� . . . . 170

11.5.1 HW ½ðQp=ZpÞ;Zp; ðQp=ZpÞ� as a Locally Compactand Totally Disconnected Topological Group . . . . . . . 170

11.6 Wigner and Weyl Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 17311.7 The Complete Chain of Subsystems of R½Zp; ðQp=ZpÞ� . . . . . . 178References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

12 A Quantum System with Positions in the Profinite Group bZ . . . . . 181

12.1 The System R½bZ; ðQ=ZÞ� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18112.2 Fourier Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

12.2.1 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 184

12.3 The Heisenberg-Weyl Group HW ½ðQ=ZÞ; bZ; ðQ=ZÞ� . . . . . . . . 185

12.3.1 HW ½ðQ=ZÞ; bZ; ðQ=ZÞ� as a Locally Compactand Totally Disconnected Topological Group . . . . . . . 186

12.4 Wigner and Weyl Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 18812.5 The Directed-Complete Partial Order of Subsystems

of R½bZ; ðQ=ZÞ�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19212.6 Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

x Contents

Mathematical Symbols

R Real numbersQ Rational numbersZ IntegersZþ or N Natural numbersZþ0 Positive integers and zero

R=Z Real numbers on a circleZðdÞ Integers modulo d½ZðdÞ�� Group of reduced residue classes modulo d (in-

vertible elements of ZðdÞ)uðdÞ Euler's totient functionJkðdÞ Jordan totient functionwðdÞ Dedekind psi functionP Set of prime numbers� Partial order (the same symbol is used for different

partial orders, e.g., divisor, subset, subgroup, andsubsystem.)

kjn or k � n k is a divisor of n (the partial order is divisibility)/ Normal subgroupPðkÞ Set of prime numbers which are divisors of kGCDðq; rÞ Greatest common divisorNS The directed complete partial order of supernatural

(Steinitz) numbersNSðpÞ The complete chain fp; p2; . . .; p1g�

Qp2P p1 (the supremum element of NS)

EQ

p2P p (element of NS)eG Pontryagin dual group to GCðdÞ ffi ZðdÞ Pontryagin dual group to ZðdÞGFðpeÞ Galois fieldR½ZðdÞ� Quantum system with positions and momenta in

ZðdÞ

xi

xdðaÞ ¼ xðaÞ exp i 2pad� �

; a 2 ZðdÞH½ZðdÞ� d-dimensional Hilbert space of the system R½ZðdÞ�jX;mi; m 2 ZðdÞ Position states in H½ZðdÞ�jP;mi; m 2 ZðdÞ Momentum states in H½ZðdÞ�Dða; bÞ; a; b 2 ZðdÞ Displacement operators for R½ZðdÞ�Pða; bÞ; a; b 2 ZðdÞ Parity operators for R½ZðdÞ�HW ½ZðdÞ� Heisenberg–Weyl group for the system R½ZðdÞ�jC; a; bi; a; b 2 ZðdÞ Coherent states in H½ZðdÞ� (C indicates coherent

states)Sðj; kjl; mÞ j; k; l; m 2 ZðdÞ Symplectic transformations for the system R½ZðdÞ�

jm� kl ¼ 1Sp½2;ZðdÞ� Symplectic group for the system R½ZðdÞ�jXðl; mÞ;mi jXðl; mÞ;mi ¼ Sðj; kjl; mÞjX;mi in H½ZðdÞ�jPðj; kÞ;mi jPðj; kÞ;mi ¼ Sðj; kjl; mÞjP;mi in H½ZðdÞ�HWSp½ZðdÞ� The group HW ½ZðdÞ�oSp½2;ZðdÞ� of displace-

ments and symplectic transformations for the sys-tem R½ZðdÞ�

Wða; b; hÞ; a; b 2 ZðdÞ Wigner function of the operator h, for R½ZðdÞ�eW ða;b; hÞ; a; b 2 ZðdÞ Weyl function of the operator h, for R½ZðdÞ�Lðq; rÞ q; r 2 ZðdÞ Line through the origin in the ZðdÞ � ZðdÞ finite

geometry (unfactorized notation)Lðm1; m2Þmi ¼ �1; . . .pi � 1

Line through the origin in the Zðp1p2Þ � Zðp1p2Þfinite geometry (factorized notation)

jXðm1; m2Þ;m1;m2imi ¼ �1; . . .pi � 1

Weak mutually unbiased bases in H½Zðp1p2Þ�(factorized notation)

^ Conjunction (logical AND)_ Disjunction (logical OR): Negation (logical NOT)" Logical NAND# Logical NORR½GFðpeÞ� Quantum system with positions and momenta in

GFðpeÞXðsÞ exp i 2pse

� �; s 2 ZðeÞ

H½GFðpeÞ� pe-dimensional Hilbert space of the R½GFðpeÞ�jX;mi; m 2 GFðpeÞ Position states in H½GFðpeÞ�jP;mi; m 2 GFðpeÞ Momentum states in H½GFðpeÞ�Dða; bÞa;b 2 GFðpeÞ Displacement operators for R½GFðpeÞ�HW ½GFðpeÞ� Heisenberg–Weyl group for R½GFðpeÞ�Pða; bÞa; b 2 GFðpeÞ Parity operators for R½GFðpeÞ�Sðj; kjl; mÞj; k; l; m 2 GFðpeÞ

Symplectic transformations for R½GFðpeÞ�jm� kl ¼ 1

Sp½2;GFðpeÞ� Symplectic group for the system R½GFðpeÞ�HWSp½GFðpeÞ� The group HW ½GFðpeÞ�oSp½2;GFðpeÞ� of dis-

placements and symplectic transformations forR½GFðpeÞ�

xii Mathematical Symbols

G Frobenius transformations for R½GFðpeÞ�Galðej1Þ Galois group of Frobenius transformations for

R½GFðpeÞ�HWGal½GFðpeÞ� The group HW ½GFðpeÞ�oGal½GFðpeÞ� of displace-

ments and Frobenius transformations for R½GFðpeÞ�SpGal½GFðpeÞ� The group Sp½2;GFðpeÞ�oGal½GFðpeÞ� of sym-

plectic and Frobenius transformations forR½GFðpeÞ�

HWSpGal½GFðpeÞ� The group HWSp½GFðpeÞ�oGal½GFðpeÞ� of dis-placements and symplectic and Frobenius transfor-mations for R½GFðpeÞ�

Qp p-adic numbersZp p-adic integersQp=Zp Fractional p-adic numbersCðp1Þ ffi Qp=Zp Prüfer groupR½Zp; ðQp=ZpÞ� Quantum system with positions in Zp and momenta

in fZp ffi Qp=Zp

B½Zp; ðQp=ZpÞ� The Schwartz–Bruhat space for R½Zp; ðQp=ZpÞ�LC½fpðapÞ� Degree of local constancy of a complex function

fpðapÞ with ap 2 Qp

CS½fpðapÞ� Degree of compact support of a complex functionfpðapÞ with ap 2 Qp

Apðk; nÞ Set of functions on Qp with CS½fpðapÞ� ¼ k andLC½fpðapÞ� ¼ n

Apðk; �ÞS

n Apðk; nÞApð�; nÞ

Sk Apðk; nÞ

ApS

k;n Apðk; nÞDpðap; bpÞap 2 Qp=Zp; bp 2 Zp

Displacement operators for R½Zp; ðQp=ZpÞ�

HW ½ðQp=ZpÞ;Zp; ðQp=ZpÞ� Heisenberg–Weyl group for the systemR½Zp; ðQp=ZpÞ�

Q=Z ffi Pp2P Qp=Zp Rational numbers on a circle

bZQ

p2P Zp

R½bZ; ðQ=ZÞ� System with positions in bZ and momenta inebZ ffi Q=Z

B½bZ; ðQ=ZÞ� The Schwartz–Bruhat space for R½bZ; ðQ=ZÞ�Dða; bÞ a 2 Q=Z; b 2 bZ Displacement operators for R½bZ; ðQ=ZÞ�HW ½ðQ=ZÞ; bZ; ðQ=ZÞ� Heisenberg–Weyl group of displacements for

R½bZ; ðQ=ZÞ�

Mathematical Symbols xiii

Chapter 1Introduction

Abstract This chapter presents the contents of this monograph, and highlights itsnovel flavour.

Quantum mechanics is usually studied for systems where both position andmomentum take values in R. However more generally, the position can take val-ues in an Abelian locally compact group G, and the momentum in its Pontryagindual group ˜G [1–3]. Pontryagin duality between two groups which are used forpositions and momenta, is the fundamental starting point for quantum mechanics(and harmonic analysis). In this monograph G is one of the finite or profinite groupsdiscussed below.

1.1 Finite Quantum Systems

We study quantum mechanics on the following finite groups:

• G ∼= ˜G = Z(d): We study a quantum system with variables in the ring Z(d),and denote it as Σ[Z(d)] [4–7]. In quantum information this system is calledqudit. Genuine examples of such systems are spins. Other examples are infinite-dimensional systems, which due to low energy operate in the lowest d states. Inthis case the infinite-dimensional system is approximatedwith a finite-dimensionalquantum system. Superconducting qubits are examples of this.In the case that d = p (where p is a prime number), Z(p) is a field and hasstronger properties than a ring. Consequently the corresponding systemsΣ[Z(p)]have stronger properties (e.g., in relation to mutually unbiased bases) which arediscussed in this monograph.

• G ∼= ˜G = GF(pe): Apart from Z(p), the Galois fields GF(pe) are also finitefields. Quantum systems with variables in GF(pe), are denoted as Σ[GF(pe)]

© Springer International Publishing AG 2017A. Vourdas, Finite and Profinite Quantum Systems, Quantum Scienceand Technology, DOI 10.1007/978-3-319-59495-8_1

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2 1 Introduction

and are called Galois quantum systems [8]. They inherit Galois structure (e.g.,Frobenius transformations and the Galois group), and have other desirable prop-erties (e.g., in relation to mutually unbiased bases).

We call the systems Σ[Z(d)] and Σ[GF(pe)], finite quantum systems.

1.2 Profinite Quantum Systems

Profinite groups are groups at the edge of very large finite groups. The concept ofinverse limit defines this in a rigorous manner. Topologically, they are totally dis-connected (an intermediate concept between discrete and continuous). We considerthe following profinite groups:

• G = Zp and ˜G = Qp/Zp: Theprofinite groupZp of p-adic integers, canbeviewedas the group Z(pe), with very large e. Its Pontryagin dual group is the group offractional p-adic numbers Qp/Zp. We study a quantum system with positionsin Zp and momenta in Qp/Zp, and denote it Σ[Zp, (Qp/Zp)] [9–11]. It can beviewed as a quantum system Σ[Z(pe)] with very large e, in the following sense.The set of systems Σ[Z(pe)] with the order ‘subsystem’, is a chain. This chain isnot complete, but it becomes complete if we add the Σ[Zp, (Qp/Zp)] to it.

• G = ̂Z and ˜G = Q/Z: The profinite group ̂Z, can be viewed as the group Z(d),with very large d. Its Pontryagin dual group is the groupQ/Z of rational numberson a circle. We study a quantum system with positions in ̂Z and momenta inQ/Z, and denote it Σ[̂Z, (Q/Z)] [9–11]. It can be viewed as a quantum systemwith variables in Σ[Z(d)] with very large d, in the following sense. The set of thesystemsΣ[Z(d)]with the order ‘subsystem’, is a directed partial order. It becomesa directed-complete partial order, if we add theΣ[̂Z, (Q/Z)] (and also some othersystems as discussed later) to it.

We call the Σ[Zp, (Qp/Zp)] and Σ[̂Z, (Q/Z)], profinite quantum systems.

1.3 Brief Overview

We describe briefly the content of each chapter.In Chap.2, we present some background material on partial orders. In particular

we consider the set N of natural numbers, which with the order divisibility, is adirected partial order. We add ‘top elements’ to it, and extend it into the set ofsupernatural (Steinitz) numbers, which is a directed-complete partial order. We alsopresent briefly some aspects of Pontryagin duality.

In Chap.3, we discuss the ringZ(d) and its factorization in terms of smaller rings,which is based on the Chinese remainder theorem. We also discuss the symplecticgroup Sp[2,Z(d)] and its factorization in terms of smaller symplectic groups basedon the Chinese remainder theorem.

1.3 Brief Overview 3

In Chap.4, we study quantum systemsΣ[Z(d)], with variables inZ(d). There aretechnical differences in the two cases that d is odd or even integer. In this monographwe consider the case that d is an odd integer.We discuss various phase spacemethodsin this context. The phase space is the toroidal lattice Z(d) × Z(d), and in it westudy displacements and the Heisenberg-Weyl group HW [Z(d)], and symplectictransformations and the Sp[2,Z(d)] group. We also study coherent states, Wignerfunctions, and Weyl functions, in this context. Using the factorization of the ringZ(d) in terms of smaller rings, the system Σ[Z(d)] is factorized in terms of smallersystems. An analytic representation of the quantum states, that uses Theta functions,is also discussed.

In Chap.5, we discuss the phase space Z(d) × Z(d) as a finite geometry [12–14].We also discuss mutually unbiased bases [15–17]. They have important applicationsin quantum communications and quantum cryptography, and their study involvesdeep problems in discrete mathematics. We consider the following cases:

• For d = p (where p is a prime number),Z(p) × Z(p) is a near-linear finite geom-etry (two lines have at most one point in common). In this case, the number ofmutually unbiased bases is p + 1, and there is a duality between them and thep + 1 lines through the origin in Z(p) × Z(p).

• For d = pe (a power of prime number), the discussion on mutually unbiased basesinvolves the Galois field GF(pe), and is presented later in Chap.9.

• For d which is not a prime number or a power of prime number, it is difficult to findthe maximum number of mutually unbiased bases. In this case the Z(d) × Z(d)

is a non-near-linear finite geometry (two lines have many points in common).Motivated by a desire to have a duality between bases and lines, we discuss aweaker concept which we call weak mutually unbiased bases [18–20].

Overall, Chap. 5 discusses mutually unbiased bases and their generalizations, andtheir duality to lines in the finite geometry Z(d) × Z(d).

In Chap.6, we discuss the quantum logic of finite quantum systems. The discus-sion starts with Boolean algebras and Boolean rings describing classical logic, andclassical computation. We then discuss the Birkhoff-von Neumann lattice [21–25],which describes the quantum logic, and which for systems with finite-dimensionalHilbert space, is a modular orthocomplemented lattice.

In Chap.7, we discuss applications of the formalism of finite quantum systems.They include angle and angular momentum states, interferometry of multimode sys-tems, orbital angular momentum states, etc. Each of these topics is a subject in itsown right, and the intention is to define the basic quantities and provide a guide tothe literature.

In Chap.8, we present background material from Galois theory, which is neededlater. The emphasis is on how to do practical calculations with Galois numbers.

InChap.9,wediscuss theGalois quantumsystemsΣ[GF(pe)].Manyof the prop-erties of these systems are similar to those of the finite quantum systems Σ[Z(d)].The emphasis in our presentation is on extra ‘Galois properties’ that they have, likethe Frobenius transformations and the Galois group. We also discuss mutually unbi-ased bases in these systems.

4 1 Introduction

In Chap.10, we present background material from p-adic numbers and profinitegroups, which is needed later. We explain that the Pontryagin dual group to Zp is theQp/Zp, and the Pontryagin dual group to ̂Z is Q/Z. The inverse and direct limitsare central concepts, for these discussions. The emphasis is on how to do practicalcalculations with p-adic numbers.

In Chap.11, we discuss the quantum system Σ[Zp, (Qp/Zp)]. Using the the-ory of profinite groups and inverse and direct limits, we show rigorously that theΣ[Zp, (Qp/Zp)] can be viewed as the system Σ[Z(pe)] in the limit of large e. Wediscuss the concepts of locally constant functions, and functions with constant sup-port, which are designed to make integrals to converge. We also define the Schwartz-Bruhat space B[Zp, (Qp/Zp)] of wavefunctions, and then Fourier transforms, theHeisenberg-Weyl group HW [(Qp/Zp),Zp, (Qp/Zp)], and Wigner and Weyl func-tions. It is shown that the HW [(Qp/Zp),Zp, (Qp/Zp)] is a totally disconnected andlocally compact topological group.

In Chap.12, we discuss the quantum system Σ[̂Z, (Q/Z)]. Using the theory ofprofinite groups and inverse and direct limits, we show that the Σ[̂Z, (Q/Z)] can beviewed as the system Σ[Z(d)] in the limit of large d. We show that the Schwartz-Bruhat spaceB[̂Z, (Q/Z)] of wavefunctions is the restricted tensor product of all theSchwartz-Bruhat spacesB[Zp, (Qp/Zp)]with all prime p (this is the tensor productwith an extra restriction). We also discuss Fourier transforms, the Heisenberg-Weylgroup HW [(Q/Z),̂Z, (Q/Z)], and Wigner and Weyl functions. It is shown thatthe HW [(Q/Z),̂Z, (Q/Z)] is the restricted direct product of the Heisenberg-Weylgroups HW [(Qp/Zp),Zp, (Qp/Zp)] with all prime p (this is the direct productwith an extra restriction). The HW [(Q/Z),̂Z, (Q/Z)] is a totally disconnected andlocally compact topological group.

1.4 Aim of the Monograph

The aim of the monograph, is to present the above described material, by adding anovel flavor to it. In addition to the standard material, we emphasize some topicswhich either have not been discussed extensively in the literature, or they have beendiscussed from a different angle to the one presented here. The choice of such topicsis of course subjective, and we briefly highlight them below, firstly in the context ofQuantumMechanics and Quantum Information [26–28] and secondly in the contextof p-adic Physics [10, 11, 29–31].

1. Quantum Mechanics and Quantum Information:

• Within the formalism of qudits, we highlight the following:– The study of coherent states in finite quantum systems, and generalizationslike coherent density matrices, and coherent projectors of rank n.

– The factorization of the quantum systemΣ[Z(d)] in terms of smaller quan-tum systems. This is based on the Chinese remainder theorem.

1.4 Aim of the Monograph 5

– The study of the Z(d) × Z(d) phase space, as a finite geometry. It is a near-linear finite geometrywhen d is a prime number, and a non-near-linear finitegeometry when d is not a prime number.

– Within the formalism of mutually unbiased bases, which has been studiedextensively in the literature, we also discuss a weaker concept called weakmutually unbiased bases, and emphasize their duality to the finite geometryof the phase space.

• Finite quantum systems with variables in Galois fields, have been used inmutually unbiased bases for systems with power of prime dimension. Herewe study them as a subject in its own right, with emphasis on Frobeniustransformations and the Galois group, which are the pillars of the Galoisstructure. Groups that combine Frobenius transformations with displacementsand symplectic transformations, are also discussed.

• We discuss quantum logic for finite quantum systems, and compare and con-trast it with classical logic. Logic is the heart of (classical and quantum)computer science, and it provides a language for the algebraic description ofcircuits with classical and quantum gates.

2. p-adic Physics: p-adic numbers have been used in various branches of Physics:string theory, condensedmatter, etc. In Chaps. 10, 11 and 12we present a rigorousapproach to the study of finite quantum systems, as the dimension goes to infinity.This part of the monograph belongs to the subject of ‘p-adic Physics’, but weapproach this area from a different angle, using the theory of profinite groupsand inverse and direct limits. The concept ‘profinite’ is extended from groups toquantum systems. We show that:

• The set of all Σ[Z(pe)] with e ∈ N, is a chain which is not complete. Itbecomes complete if we add to it, the Σ[Zp, (Qp/Zp)]. All the systemsΣ[Z(pe)] with e ∈ N, are subsystems of Σ[Zp, (Qp/Zp)]. In this sense,Σ[Zp, (Qp/Zp)] can be viewed as the Σ[Z(pe)] in the limit of large e.

• The set of all Σ[Z(d)] with d ∈ N is a directed partially ordered set, whichis not complete. It becomes complete if we add to it the Σ[̂Z, (Q/Z)](and also some other systems). All systems Σ[Z(d)] are subsystems of theΣ[̂Z, (Q/Z)]. In this sense, Σ[̂Z, (Q/Z)] can be viewed as the Σ[Z(d)] inthe limit of large d.

References

1. Pontryagin, L. S. (1966). Topological groups. New York: Gordon and Breach.2. Ribes, L., & Zalesskii, P. (2000). Profinite groups. Berlin: Springer.3. Wilson, J. (1998). Profinite groups. Oxford: Clarendon.4. Weyl, H. (1950). Theory of groups and quantum mechanics. New York: Dover.5. Schwinger, J. (1960). Proceedings of the National Academy of Science of the United States of

America, 46, 570.6. Schwinger, J. (1970). Quantum kinematics and dynamics. New York: Benjamin.

6 1 Introduction

7. Vourdas, A. (2004). Reports on Progress in Physics, 67, 267.8. Vourdas, A. (2007). Journal of Physics A, 40, R285.9. Gel’fand, I. M., Graev, M. I., & Piatetskii-Shapiro, I. I. (1990). Representation theory and

automorphic functions. London: Academic.10. Vladimirov, V. S., Volovich, I. V., & Zelonov, E. I. (1994). p-adic analysis and mathematical

physics. Singapore: World Scientific.11. Vourdas, A. (2013). Journal of Physics A, 46, 043001.12. Batten, L. M. (1997). Combinatorics of finite geometries. Cambridge: Cambridge University

Press.13. Hirchfeld, J. W. P. (1979). Projective geometries over finite fields. Oxford: Oxford University

Press.14. Hirchfeld, J.W. P.,&Thas, J. A. (1991).GeneralGalois geometries. Oxford:OxfordUniversity

Press.15. Wootters, W., & Fields, B. D. (1989). Annals of Physics (NY), 191, 363.16. Kibler, M. (2009). Journal of Physics A, 42, 353001.17. Durt, T., Englert, B. G., Bengtsson, I., & Zyczkowski, K. (2010). International Journal of

Quantum Computation, 8, 535.18. Shalaby, M., & Vourdas, A. (2012). Journal of Physics A, 45(052001), 1–15.19. Shalaby, M., & Vourdas, A. (2013). Mutually unbiased projectors and duality between lines

and bases in finite quantum systems. Annals of Physics, 337, 208.20. Olupitan, T., Lei, C., & Vourdas, A. (2016). An analytic function approach to weak mutually

unbiased bases. Annals of Physics, 371, 1.21. Birkhoff, G., & von Neumann, J. (1936). Annals of Mathematics, 37, 823.22. Mackey,G.W. (1963).Mathematical foundations of quantummechanics. NewYork:Benjamin.23. Piron, C. (1976). Foundations of quantum physics. New York: Benjamin.24. Jauch, J. (1968). Foundations of quantum mechanics. Reading: Addison-Wesley25. Beltrametti, E., Cassinelli, G. (1981). The logic of quantum mechanics. Reading: Addison-

Wesley26. Nielsen, M., & Chuang, I. L. (2000). Quantum computation and quantum information. Cam-

bridge: Cambridge University Press.27. Kitaev, A Yu., Shen, A. H., & Vyalyi, M. N. (2002). Classical and quantum computation.

Province: American Mathematical Society.28. Kaye, P., Laflamme, R., & Moscha, M. (2006). Introduction to quantum computing. Oxford:

Oxford University Press.29. Rammal, R., Toulouse, G., & Virasoro, M. A. (1986). Reviews of Modern Physics, 58, 765.30. Brekke, L., & Freund, P. (1993). Physics Reports, 233, 1.31. Albeverio, S., Khrennikov, A., & Shelkovich, V. M. (2010). Theory of p-adic distributions.

Cambridge U. P: Cambridge.

Chapter 2Partial Orders and Pontryagin Duality

Abstract Partial orders, supernatural numbers, and Pontryagin duality, are dis-cussed.

2.1 Partial Orders

Definition 2.1 (i) A partially ordered set is a set S with a relation ≺, and theproperties:

(1) (reflexivity) a ≺ a, for all a ∈ S;(2) (antisymmetry) if a ≺ b and b ≺ a, then a = b;(3) (transitivity) if a ≺ b and b ≺ c, then a ≺ c.

(ii) A directed partially ordered set S, is a partially ordered set with the additionalproperty that for a, b ∈ S, there exists c ∈ S such that a ≺ c and b ≺ c.

a, b are comparable if a ≺ b or b ≺ a. A partially ordered set where any pair ofelements is comparable, is a chain (total order).

Definition 2.2 Two partially ordered sets (S,≺) and (S′,≺′) are order isomorphic,if there is a bijective map f from S to S′, and f (a1) ≺′ f (a2), if and only if a1 ≺ a2.

Example 2.1

• the partial order ‘subgroup’ in a set of groups• the partial order ‘less or equal’ in the set of natural numbersN (i.e., a ≺ b if a ≤ b)• the partial order ‘divisibility’ in the set of natural numbers N (i.e., a ≺ b if a|b)For simplicity we use the same symbol ≺ for different partial orders, and its precisemeaning is clear from the context.

Definition 2.3 An upper bound of a subset T of the partially ordered set S, is anelement a ∈ S such that b ≺ a for all b ∈ T . If the set of all upper bounds of T hasa smallest element, it is called the supremum of T .

© Springer International Publishing AG 2017A. Vourdas, Finite and Profinite Quantum Systems, Quantum Scienceand Technology, DOI 10.1007/978-3-319-59495-8_2

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