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Aspects of M-Theory and Quantum Information Imperial College London Department of Theoretical Physics Aspects of M-theory and Quantum Information Leron Borsten September, 2010 Supervised by Professor M. J. Duff FRS Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Theoretical Physics of Imperial College London and the Diploma of Imperial College London 1 Declaration I herewith certify that all material in this dissertation which is not my own work has been properly acknowledged. Leron Borsten 3 4 Abstract As the frontiers of physics steadily progress into the 21st century we should bear in mind that the conceptual edifice of 20th-century physics has at its foundations two mutually incompatible theories; quantum mechanics and Einstein’s general theory of relativity. While general relativity refuses to succumb to quantum rule, black holes are raising quandaries that strike at the very heart of quantum theory. M-theory is a compelling candidate theory of quantum gravity. Living in eleven dimensions it encompasses and connects the five possible 10-dimensional superstring theories. However, M- theory is fundamentally non-perturbative and consequently remains largely mysterious, offering up only disparate corners of its full structure. The physics of black holes has occupied centre stage in uncovering its non-perturbative structure. The dawn of the 21st-century has also played witness to the birth of the information age and with it the world of quantum information science. At its heart lies the phenomenon of quantum entanglement. Entanglement has applications in the emerging technologies of quantum computing and quantum cryptography, and has been used to realize quantum teleportation experimentally. The longest standing open problem in quantum information is the proper characterisation of multipartite entanglement. It is of utmost importance from both a foundational and a technological perspective. In 2006 the entropy formula for a particular 8-charge black hole appearing in M-theory was found to be given by the ’hyperdeterminant’, a quantity introduced by the mathematician Cayley in 1845. Remarkably, the hyperdeterminant also measures the degree of tripartite entanglement shared by three qubits, the basic units of quantum information. It turned out that the different possible types of three-qubit entanglement corresponded directly to the different possible subclasses of this particular black hole. This initial observation provided a link relating various black holes and quantum infor- mation systems. Since then, we have been examining this two-way dictionary between black holes and qubits and have used our knowledge of M-theory to discover new things about multipartite en- tanglement and quantum information theory and, vice-versa, to garner new insights into black holes and M-theory. There is now a growing dictionary, which translates a variety of phenomena in one language to those in the other. Developing these fascinating relationships, exploiting them to better understand both M-theory and quantum entanglement is the goal of this thesis. In particular, we adopt the elegant mathematics of octonions, Jordan algebras and the Freudenthal triple system as our guiding framework. In the course of this investigation we will see how these fascinating algebraic structures can be used to quantify entanglement and define new black hole dualities. 5 6 Acknowledgments First and foremost, I would like to thank my supervisor Michael Duff for his continual support, guidance, patience, encouragement, insight and especially for his generosity with both his time and ideas. I couldn’t have hoped for a better supervisor. It has been a real pleasure talking physics, football and much else besides! I would also very much like to thank my PhD student collaborators and great friends: Duminda Dahanayake, Hajar Ebrahim and William Rubens. It has been a real hoot working with you chaps. We even did some physics. I am also very grateful to Duminda for the diagrams appearing in this thesis. A big thank you to all my fellow PhD students for making the last four years so much fun. In particular, I would like to thank Steven Johnston, Lydia Philpott and Ben Withers (especially for the interesting travel experiences we shared). To all of the theoretical physics group at Imperial, thank you for making it such a great place to work. A special thanks to Graziela De Nadai-Sowrey, Fay Dowker, Jerome Gauntlett, Chris Isham and Daniel Waldram for all your help and advice, scientific and otherwise. Thank you to Chris for introducing me to theoretical physics. I would like to extend my gratitude to our collaborators Sergio Ferrara and Alessio Marrani, it has been a most enjoyable and enlightening experience working with you both. I would like to thank Sergio Ferrara for giving me the opportunity to visit CERN. I would also like to thank my PhD examiners Fay Dowker and Peter West. Finally, I would like to thank my family and friends, life, never mind the PhD, is not possible without you lot. A very special thanks to my parents for their unwavering practical and emotional support. All my love to Narmadha. 7 8 Welcome, O life! I go to encounter for the millionth time the reality of experience and to forge in the smithy of my soul the uncreated conscience of my race 9 10 Contents 1 Introduction 17 1 Overview . 17 2 Structure of thesis . 24 I BLACK HOLES AND QUANTUM INFORMATION 27 2 Quantum information and entanglement 29 1 A brief introduction to quantum information . 31 1.1 Qubits . 32 1.2 Entanglement and the Bell inequality . 34 1.3 Entanglement dependent quantum information . 37 2 Entanglement classification . 40 2.1 Bell inequalities without the inequality . 40 2.2 The SLOCC paradigm . 42 2.3 Entanglement measures . 43 2.4 Stochastic LOCC equivalence . 44 2.5 Two qubit entanglement . 46 2.6 Three qubits . 48 3 M-theory and black holes 55 1 The road to M-theory . 56 2 U-duality . 58 3 Black holes in supergravity . 61 4 The ST U model . 64 4 Black holes and qubits 67 1 The ST U model and tripartite entanglement of three qubits . 67 1.1 Entropy/entanglement correspondence . 67 1.2 Classification of N = 2 black holes and three-qubit states . 68 1.3 Higher order corrections . 71 1.4 Attractors and SLOCC . 72 2 N = 8 supergravity and the tripartite entanglement of seven qubits . 73 2.1 N = 8 supergravity and black holes . 74 3 E7 and the tripartite entanglement of seven qubits . 76 11 4 Classification of N = 8 black holes and seven-qubit states . 79 5 Further developments . 80 5.1 Microscopic interpretation . 80 5.2 Supersymmetric quantum information . 81 5.3 4-qubit entanglement and the ST U model in D = 3 . 82 II ALGEBRAIC PERSPECTIVES 83 5 Algebras 85 1 Composition algebras . 85 1.1 Octonions . 87 2 Jordan algebras . 89 2.1 Preliminaries . 89 2.2 Cubic Jordan algebras . 90 2.3 Cubic Jordan algebras: Examples . 92 3 The Freudenthal triple system . 96 3.1 Axiomatic definition . 96 3.2 Definition over a Jordan Algebra . 97 3.3 Symmetries . 98 3.4 FTS: Examples . 102 6 Algebraic black p-branes 108 1 Jordan algebras and black holes (strings) in D = 5 . 109 1.1 Jordan algebras and N = 2 Maxwell-Einstein supergravities . 110 1.2 Jordan algebras and N = 8 supergravity . 112 2 Freudenthal triple systems and black holes in D = 4 . 114 2.1 The FTS and N = 2 Maxwell-Einstein supergravities . 115 7 N = 8 supergravity and the tripartite entanglement of seven qubits: revisited 121 1 The Fano plane, octonions and the tripartite entanglement of seven qubits . 122 1.1 Subsectors . 123 1.2 Discrete symmetry of the fano plane . 126 1.3 Three descriptions . 129 8 The FTS classification of qubit entanglement 137 1 The FTS classification of three qubit entanglement . 137 1.1 The ST U Freudenthal triple system . 137 1.2 The 3-qubit Freudenthal triple system . 139 1.3 The FTS rank entanglement classes . 140 1.4 SLOCC orbits . 142 2 Generalising to an n-qubit FTS . 142 2.1 Three qubits and the FTS re-examined . 143 2.2 Examples . 148 2.3 Further work . 151 12 9 Integral structures 155 1 Integral algebras and their symmetries . 155 1.1 The integral split-octonions . 155 1.2 Integral Jordan algebras . 157 1.3 Integral Freudenthal triple system . 158 2 Integral U-duality orbits . 160 2.1 D = 6; N = 8 black string charge orbits . 161 2.2 D = 5; N = 8 black hole charge orbits . 163 2.3 D = 4; N = 8 black hole charge orbits . 164 2.4 Conclusion . 168 3 Freudenthal duality . 170 3.1 Introduction . 170 3.2 The 4D Freudenthal dual . 171 3.3 The action of F-duality on arithmetic U-duality invariants . 172 3.4 F-dual in canonical basis . 175 3.5 The NS-NS sector . 177 3.6 The 5D Jordan dual . 179 3.7.
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