Lecture Notes in Physics

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1 Lecture Notes in Physics Founding Editors: W. Beiglböck, J. Ehlers, K. Hepp, H. Weidenmüller Editorial Board R. Beig, Vienna, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France F. Guinea, Madrid, Spain P. Hänggi, Augsburg, Germany W. Hillebrandt, Garching, Germany R. L. Jaffe, Cambridge, MA, USA W. Janke, Leipzig, Germany H. v. Löhneysen, Karlsruhe, Germany M. Mangano, Geneva, Switzerland J.-M. Raimond, Paris, France M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany J. Zittartz, Köln, Germany

2 The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: to be a compact and modern up-to-date source of reference on a well-defined topic to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse Heidelberg / Germany christian.caron@springer.com

3 F. Benatti M. Fannes R. Floreanini D. Petritis (Eds.) Quantum Information, Computation and Cryptography An Introductory Survey of Theory, Technology and Experiments 123

4 Fabio Benatti Università Trieste Dipto. Fisica Teorica Strada Costiera, Trieste Miramare Italy Roberto Floreanini Dipartimento di Fisica Teorica Strada Costiera 11 I Trieste Italy Mark Fannes Afdeling Theoretische Fysica Celestijnenlaan 200d B-3001 Heverlee Belgium Dimitri Petritis Institut de Recherche Mathématique de Rennes Université de Rennes 1 Campus de Beaulieu Rennes Cedex France Dimitri.Petritis@univ-rennes1.fr Benatti F. et al. (Eds.): Quantum Information, Computation and Cryptography: An Introductory Survey of Theory, Technology and Experiments, Lect. Notes Phys. 808 (Springer, Berlin Heidelberg 2010), DOI / Lecture Notes in Physics ISSN e-issn ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Integra Software Services Pvt. Ltd., Pondicherry Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface This book is intended for undergraduate students with a minimal physical and mathematical background; its purpose is to provide them with an introduction to the basic tools of the theory and technology of quantum information theory which comprehends quantum information proper, quantum communication, quantum computation, and quantum cryptography. The structure and contents of the book have been suggested by the following two observations: the rich interdisciplinary context that has resulted from 20 years of joint efforts of researchers from as different fields as quantum mechanics, both theoretical and experimental, classical information theory, computer science, and cryptography; the lack of a textbook for undergraduate students that need to be guided step by step through their many interconnections; indeed, the many existing books address readers already actively researching in these fields. Complying with the above two points, the book consists of the integrated contributions of various experts and must not be taken as an overview of the results so far obtained in their fields, rather as a textbook specifically thought for the class of readers specified above. With this in mind, all the authors have concentrated on those aspects they deemed important for an undergraduate student to be instructed about. An effort has thus been made to clarify with theory, examples and practical applications a series of carefully selected and correlated aspects of the theory, rather than trying to comprehensively cover the whole of quantum information theory. The book benefits from the past and present scientific activity of the authors both as researchers and as teachers and from their personal viewpoints on the subjects treated in the chapters. In particular, some important issues are approached from different and stimulating perspectives that may only improve their comprehension. Also, an extra flavor comes to the book from the participation of some of the authors in the construction of what was known as quantum probability, a theory which predated quantum information and has provided the latter with useful technical tools and ideas. Their chapters reflect the authors long acquaintance with the intriguing relations between quantum mechanics and probability, thus offering an opportunity to fully appreciate the subsequent developments and achievements. The book consists of ten chapters; the first one by D. Petz, provides an introduction to the mathematical tools of quantum mechanics including the von Neumann v

6 vi Preface entropy and the quantum relative entropy. The second one by Y. Suhov is an introduction to classical probability and information theory, with particular emphasis on Shannon s coding theorems. In chapter Quantum Probability and Quantum Information Theory H. Maassen offers an introduction to the physics of quantum mechanics with particular attention to its most puzzling aspects as entanglement and the Bell s inequalities. The presentation of the quantum entanglement phenomenon is further developed by F. Benatti in chapter Bipartite Quantum Entanglement with the use of entropic tools and techniques from quantum open system theory. While chapter Quantum Probability and Information Theory and Bipartite Quantum Entanglement mainly treat the quantum mechanics of finite-level systems, in chapter Field-Theoretical Methods by R. Alicki concentrates instead on the quantum mechanics of systems with infinitely many degrees of freedom. Quantum statistical mechanics and quantum field theory are indeed becoming more and more important in the recent developments of quantum information both theoretically and experimentally. Quantum information transmission with the quantum versions of Shannon s classical coding theorems is the theme of chapter Quantum Entropy and Information by N. Datta, while chapter Photonic Realization of Quantum Information Protocols by M. Genovese and chapter Physical Realizations of Quantum Information by F. De Melo and A. Buchleitner deal with the experimental achievements in photonics, respectively, atomic physics that turned the puzzling aspects of quantum mechanics into actual physical resources; finally, chapter Quantum Cryptography by D. Bruss and T. Meyer and the last chapter by J. Kempe and T. Widick address those applications of quantum mechanics to cryptography, respectively, computation theory that so greatly contributed to the ever growing interest in the theoretical and technological issues presented in this book. All chapters contain examples, problems, and exercises whose aim is to make the students actively interact with the text. The references are provided not with the purpose of being exhaustive, an almost impossible task that is much better accomplished by the many existing advanced books and reviews. Rather, in view of the specific class of readers addressed by this book, those references to the literature have been selected that may help to integrate the authors viewpoints and to suggest the reader a path toward the latest advances of quantum information theory.

7 Contents Hilbert Space Methods for Quantum Mechanics... 1 D. Petz 1 Hilbert Spaces Postulates of Quantum Mechanics SomeApplications References Classical Information Theory Y. Suhov 1 Entropy Source Coding Channel Coding Bibliographical Notes References Quantum Probability and Quantum Information Theory H. Maassen 1 Introduction Why Classical Probability Does Not Suffice Toward a Mathematical Model Quantum Probability Operations on Probability Spaces Quantum Impossibilities Quantum Novelties References Bipartite Quantum Entanglement F. Benatti 1 Introduction Bipartite Entanglement Entanglement Detection Complete Positivity, Open Quantum Systems, and Entanglement References vii

8 viii Contents Field-Theoretical Methods R. Alicki 1 Introduction The Quantum Harmonic Oscillator Quantum Bosonic Fields Coherent and Thermal States for Bosons Second Quantization of Fermions Further Reading References Quantum Entropy and Information Nilanjana Datta 1 Introduction Preliminaries Rudiments of Classical Information Theory Quantum Entropy Data Compression in Quantum Information Theory Quantum Channels Accessible Information and the Holevo Bound References Photonic Realization of Quantum Information Protocols M. Genovese 1 Introduction Photon Entanglement Optical Realizations of Quantum Information Protocols Optical Quantum Computation Protocols Quantum Communication References Physical Realizations of Quantum Information F. de Melo and A. Buchleitner 1 Introduction ASingleQubitinInteractionwiththeRadiationField Qubit Entanglement Through the Jaynes Cummings Interaction References Quantum Cryptography D. Bruß and T. Meyer 1 Introduction ClassicalCryptography Quantum Cryptography Eavesdropping Strategies Unconditional Security of BB

9 Contents ix 6 Defense Against Eavesdropping with Photon Number Splitting (PNS) Attacks Classical Upper Bounds on the Secret Key Rate The Role of Entanglement in QKD Problems/Exercises References Quantum Algorithms J. Kempe and T. Vidick 1 Notations The Quantum Circuit Model FirstAlgorithms The Quantum Fourier Transform Deutsch Josza and Simon s Algorithms Factoring in Polynomial Time The Hidden Subgroup Problem Grover s Algorithm for Unstructured Search Developments References Index...343