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Université De Montréal Quantum Pseudo-Telepathy Games Aune ///. 3 Université de Montréal Quantum Pseudo-Telepathy Games Aune Lise Broadbent Département d’informatiqlle et de recherche opérationnelle faculté des arts et des sciences Mémoire présenté à la faculté des études supérieures en vue de l’obtention du grade de Maître ès sciences (M.Sc.) en informatique avril, 2004 \\ 07 OCT © Anne Lise Broadbent, 2004. 05Lt cl \} .c C Université dl1 de Montréal Direction des bibliothèques AVIS L’auteur a autorisé l’Université de Montréal à reproduire et diffuser, en totalité ou en partie, par quelque moyen que ce soit et sur quelque support que ce soit, et exclusivement à des fins non lucratives d’enseignement et de recherche, des copies de ce mémoire ou de cette thèse. L’auteur et les coauteurs le cas échéant conservent la propriété du droit d’auteur et des droits moraux qui protègent ce document. Ni la thèse ou le mémoire, ni des extraits substantiels de ce document, ne doivent être imprimés ou autrement reproduits sans l’autorisation de l’auteur. Afin de se conformer à la Loi canadienne sut la protection des renseignements personnels, quelques formulaires secondaires, coordonnées ou signatures intégrées au texte ont pu être enlevés de ce document. Bien que cela ait pu affecter la pagination, il n’y a aucun contenu manquant. NOTICE The author of this thesis or dissertation has granted a nonexclusive license allowing Université de Montréal to reproduce and publish the document, in part or in whole, and in any format, solely for noncommercial educational and research purposes. The author and co-authors if applicable retain copyright ownership and moral rights in this document. Neither the whole thesis or dissertation, flot substantial extracts from it, may be printed or otherwise reproduced without the author’s permission. In compliance with the Canadian Privacy Act some supporting forms, contact information or signatures may have been removed from the document. While this may affect the document page count, it does flot represent any loss of content from the document. Université de Montréal faculté des études supérieures Ce mémoire intitulé: Quantum Pseudo-Telepathy Games présenté par: Aune Lise Broadbent a été évalué par un jury composé des personnes suivantes: Michel Boyer président-rapporteur Alain Tapp directeur de recherche Gilles Brassard codirecteur Stefan Wolf membre du jury Mémoire accepté le RÉSUMÉ Le traitement de l’information quantique est au confluent des sciences physique, mathématiques et informatique; il vise à déterminier ce qu’on peut et e peut pas faire avec l’information quantique. Le sujet de ce mémoire est la complexité de la communication, qui est un domaine de l’informatique qui vise la quantification de la communication nécessaire à la résolution de problèmes distribués. La pseudo-télépathie est une application surprenante du traitement de l’infor mation quantique à la complexité de la communication. Grâce à une ressource quantique appelée « intrication », deux joueurs ou plus peuvent accomplir une tâche sans comrnnnïqner, tandis que ceci serait impossible pour des joueurs clas siques (qui n’ont pas accès à l’intrication). Un jeu de pseudo-télépathie à n joueurs se présente comme suit: chaque joueur reçoit en entrée une question. Sans com muniquer, chacun émet en sortie une réponse. Le jeu est gagné si les réponses conjointes satisfont une certaine condition. Il s’agit d’un jeu de pseudo-télépathie si les joueurs quantiques peuvent gagner de façon systématiqile, tandis que ceci est impossible pour les joueurs classiques. Dans ce mémoire, nous décrivons sept jeux de pseudo-télépathie, tirés de la littérature de la physique et de l’informatique quantique. Nous incluons aussi des résultats originaux de l’auteur. Les jeux sont présentés du point de vue informa tique, et de façon uniforme, ce qui facilite leur comparaison. Certains points de comparaison sont: le nombre de joueurs, la taille de l’entrée, la taille de la sortie, la condition gagnante, l’état intriqué partagé et la probabilité maximale de réussite pour les joueurs classiques. Mots clés: informatique quantique, complexité de la communication quantique, non-localité, intrication, théorème de Beil, échappatoire de la détection. ABSTRACT Quantum information processing is at the crossroads of physics, mathematics and computer science; it is concerned with what we can and cannot do with quantum information. This thesis deals with communication cornplexfty, which is an area of computer science that aims at quantifying the amount of communication necessary to solve distributed problems. Pseudo-telepathy is a surprising application of quantum information processing to communication complexity. Thanks to a quantum resource called “entangle ment”, two or more quantum players can accomplish a task with no communica tion. whereas this would be impossible for classical players (who do not have access to entanglement). A pseudo-telepathy game with n players is the following: each player receives as input a question. Without communicating, each player outputs an answer. The players win if their joint answers satisfy a certain condition. \‘Ve say that the game exhibits pseudo-telepat.hy if quantum players can svstematically succeed at this game, whereas this woiild be impossible for classical players. In this thesis, we describe seven pseudo-telepathy games which appear in the physics and quantum information processing literature. We have also included original resuits of the author. The games are presented frohi a computer scientist’s perspective, and in a uniform way, in order to facilitate comparison. Some points of comparison are: number of players, size of the inputs, size of outputs, winning condition, shared entangled state and maximum success probability for classical players. Keywords: quantum information processing, quantum communica tion complexity, nonlocality, entanglement, Beil’ s theorem, detection loophole. V X 1 3 4 5 6 7 .CONTENTS RÉSUMÉ iII ABSTRACT iv CONTENTS LIST 0F TABLES viii LIST 0F FIGURES LIST 0F APPENDICES xi LIST 0F ABBREVIATIONS xii NOTATION xiii DEDICATION xiv ACKNOWLEDGEMENTS XV PREFACE xvi CHAPTER 1: INTRODUCTION 1.1 Measurements and Spooky Action at a Distance 1.2 Bell’s Response 1.3 Pseudo-Teïepathy 1.3.1 Telepathy alld Pseudo-Telepathy . 1.3.2 Pseudo-Telepathy and Bell’s Theorem 1.4 Related Work 1.5 Contributions 1.6 Structure of the Thesis 28 31 32 34 37 41 42 43 44 vi CHAPTER 2: QUANTUM INFORMATION PROCESSING 8 2.1 The Qubit 8 2.2 Complex Tuner Product Space 9 2.3 Basic Operations 9 2.4 n-Qubit Systems 11 2.5 Operations on Parts of a System 12 2.6 Entanglement 13 CHAPTER 3: PSEUDO-TELEPATHY 15 3.1 Playing the Games 15 3.2 $trategies 17 3.2.1 The Promise 20 3.3 Physical Realizatiolls and Loopholes 22 3.3.1 Noisy Detectors 25 3.3.2 Inefficient Detectors 25 3.4 Presentation of the Games 26 CHAPTER 4: TWO-PARTY GAMES 28 4.1 The Impossible Colouring Game 4.1.1 A Quantum Winning Strategy 4.1.2 Classical Success Proportion 4.1.3 Special Case of the Impossible Colouring Game 33 4.2 The Distributed Deutsch-Jozsa Game 33 4.2.1 A Quantum Winning Strategy 4.2.2 Classical Success Proportion 4.3 The Magic Square Game 3$ 4.3.1 A Quantum Wirining $trategy 39 4.3.2 Classical Success Proportion 4.4 Cabello’s Game 4.4.1 A Quantum Winning Strategy 4.4.2 Classical Sllccess Proportion vi 4.5 The Magie Square and Cabello’s Games Are Equivalent . 44 4.6 The Matching Game 49 4.6.1 A Quantum Winning $trategy . 50 4.6.2 The Hidden Matching Problem . 52 4.6.3 Classical Success Proportion 52 CHAPTER 5: MULTI-PARTY GAMES 54 5.1 The Mermin-GHZ Three-Party Game 54 5.1.1 A Quantum Winning Strategy 55 5.1.2 Classical Success Proportion 57 5.2 The Parity Game 59 5.2.1 A Quantum Winning Strategy 59 5.2.2 Classical Success Proportion 61 5.2.3 Classical Success Probability 75 5.2.4 Towards Closing the Detection Loophole 87 5.3 The Extended Parity Game 94 5.3.1 A Quantum Winning Strategy 95 5.3.2 Classical Success Proportion 96 5.3.3 Towards Closing the Detection Loophole 97 CHAPTER 6: CONCLUSION 101 6.1 Future Work 105 BIBLIOGRAPHY 106 LIST 0F TABLES 3.1 Presentation of the games 27 4.1 Impossible colouring game 31 4.2 Impossible colouring game (rn = 3) 33 4.3 Distributed Deutsch-Jozsa game 35 4.4 Magic square game 39 4.5 A 3 x 3 array of observables 40 4.6 Alice’s strategy 42 4.7 Bob’s strategy 42 4.8 Cabello’s game 43 4.9 Winning conditions for Cabello’s game . 43 4.10 Winning conditions for the magic square and Cabello’s game . 49 4.11 Matching game 50 5.1 Mermin-GHZ game 55 5.2 Parity game 60 5.3 Number of questions that yield a winning answer 66 5.4 Values of n/2 + 3n (mod 4) for even n 74 5.5 Optimal strategies 75 5.6 Values of £ (mod 4) for odd n 81 5.7 Values of n — 21? (mod 8) for odd n 82 5.8 Values of £ (mod 4) for even n 84 5.9 Values of n — 21? (mod 8) for even n 84 5.10 Values of n — 21?3 (mod 8) for even n 85 5.11 Extended parity game 95 6.1 Two-party games: comparison table part I . 102 6.2 Two-party games: comparison table part II . 103 . ix 6.3 Multi-party games: comparison table 104 LIST 0F FIGURES 3.1 A Pseudo-Telepathy Game 16 LIST 0F APPENDICES Appendix I: Proof of lemma 5.2.5 xvii Appendix II: Proof of lemmas 5.2.12 and 5.2.13 xx LIST 0F ABBREVIATIONS BK$ Beil-Kochen-Specker EPR Einstein-Podolsky-Rosen GHZ Greenberger-Horne-Zeilinger QIP Quantum Information Processing NOTATION R real numbers C complex numbers z imaginary number, r = c complex norm of c 7d d-dimensional complex inner product space b) quantum state b) normofb) Ax) Hamming weight of a binary string x bitwise complement of x modular
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