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Quantum Information Theory and the Foundations of Quantum Mechanics

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Quantum Information Theory and the Foundations of Quantum Mechanics Quantum Information Theory and The Foundations of Quantum Mechanics Christopher Gordon Timpson The Queen’s College arXiv:quant-ph/0412063v1 8 Dec 2004 A thesis submitted for the degree of Doctor of Philosophy at the University of Oxford Trinity Term 2004 file://C:\WINDOWS\Desktop\webpage\eagles.gif Quantum Information Theory and the Foundations of Quantum Mechanics Christopher Gordon Timpson, The Queen’s College Oxford University, Trinity Term 2004 Abstract of Thesis Submitted for the Degree of Doctor of Philosophy This thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of information is to be firmly distinguished from the technical notions arising in information theory; however it is maintained that in both settings ‘information’ functions as an abstract noun, hence does not refer to a particular or substance. The popular claim ‘Information is Physical’ is assessed and it is argued that this proposition faces a destructive dilemma. Accordingly, the slogan may not be understood as an ontological claim, but at best, as a methodological one. A novel argument is provided against Dretske’s (1981) attempt to base a semantic notion of information on ideas from information theory. The function of various measures of information content for quantum systems is ex- plored and the applicability of the Shannon information in the quantum context main- tained against the challenge of Brukner and Zeilinger (2001). The phenomenon of quan- tum teleportation is then explored as a case study serving to emphasize the value of recognising the logical status of ‘information’ as an abstract noun: it is argued that the conceptual puzzles often associated with this phenomenon result from the familiar error of hypostatizing an abstract noun. The approach of Deutsch and Hayden (2000) to the questions of locality and infor- mation flow in entangled quantum systems is assessed. It is suggested that the approach suffers from an equivocation between a conservative and an ontological reading; and the differing implications of each is examined. Some results are presented on the character- ization of entanglement in the Deutsch-Hayden formalism. Part I closes with a discussion of some philosophical aspects of quantum computation. In particular, it is argued against Deutsch that the Church-Turing hypothesis is not underwritten by a physical principle, the Turing Principle. Some general morals are drawn concerning the nature of quantum information theory. In Part II, attention turns to the question of the implications of quantum information theory for our understanding of the meaning of the quantum formalism. Following some preliminary remarks, two particular information-theoretic approaches to the foundations of quantum mechanics are assessed in detail. It is argued that Zeilinger’s (1999) Founda- tional Principle is unsuccessful as a foundational principle for quantum mechanics. The information-theoretic characterization theorem of Clifton, Bub and Halvorson (2003) is assessed more favourably, but the generality of the approach is questioned and it is argued that the implications of the theorem for the traditional foundational problems in quantum mechanics remains obscure. Acknowledgements It is my pleasant duty to thank a large number of people, and more than one institution, for the various forms of help, encouragement and support that they have provided during the time I have been working on this thesis. The UK Arts and Humanities Research Board kindly supported my research with a postgraduate studentship for the two years of my BPhil degree and a subsequent two years of doctoral research. I should also like to thank the Provost and Fellows of The Queen’s College, Oxford for the many years of support that the College has provided, both material and otherwise. Reginae erunt nutrices tuae: no truer words might be said. A number of libraries have figured strongly during the time I have been at Oxford: I would like in particular to thank the staff at the Queen’s and Philosophy Faculty libraries for their help over the years. On a more personal note, I would like to extend my thanks and appreciation to my supervisor Harvey Brown, whose good example over the years has helped shape my approach to foundational questions in physics and who has taught me much of what I know. I look forward to having the opportunity in the future to continue working with, and learning from, him. Another large debt of thanks is due to John Hyman, my earliest teacher in philosophy, who has continued to offer a great deal of assistance and encouragement over the years; and whose fearsome questioning helped show me what it is to do philosophy (and, incidentally, alerted me to the dangers of pernicious theorising). Jon Barrett and I started out on the quest to understand the foundations and phi- losophy of physics at the same time, just about a decade ago, now. Since then, we have shared much camaraderie and many conversations, several of which have found their way into this thesis at one point or another. And Jon is still good enough to check my reasoning and offer expert advice. I would like to thank Jeremy Butterfield, Jeff Bub, Chris Fuchs and Antony Valentini, all of whom have been greatly encouraging and who have offered useful comments on and discussion of my work. In particular, I should single out Jos Uffink for his unstinting help in sharing his expertise in quantum mechanics, uncertainty and probability; and for providing me with a copy of his unpublished PhD dissertation on measures of uncertainty and the uncertainty principle. My understanding of measures of information has been heavily influenced by Jos’s work. The (rest of the) Oxford philosophy of physics mob are also due a great big thank- you: one couldn’t hope for a more stimulating intellectual environment to work in. So thanks especially to Katharine Brading, Guido Bacciagaluppi, Peter Morgan, Justin Pniower, Oliver Pooley, Simon Saunders and David Wallace for much fun, support and discussion (occasionally of the late-night variety). A little further afield, I would like to thank Marcus Appleby, Ari Duwell, Doreen Fraser, Hans Halvorson, Michael Hall, Leah Henderson, Clare Hewitt-Horsman (in par- ticular on the topic of Chapter 5), Richard Jozsa, James Ladyman, Owen Maroney, Michael Seevink, Mauricio Suarez, Rob Spekkens and Alastair Rae, amongst others, for stimulating conversations on information theory, quantum mechanics and physics. Finally I should like to thank my parents, Mary and Chris Timpson, sine qua non, bien sˆur; and my wife Jane for all her loving support, and her inordinate patience during the somewhat extended temporal interval over which this thesis was finally run to ground. (Oh, and she made most of the pictures too!) Contents Introduction iii I What is Information? 1 1 Concepts of Information 3 1.1 How to talk about information: Some simple ways . ..... 3 1.2 The Shannon Information and related concepts . ...... 10 1.2.1 Interpretation of the Shannon Information . ..... 10 1.2.2 Moreoncommunicationchannels. 16 1.2.3 Interlude: Abstract/concrete; technical, everyday .......... 20 1.3 AspectsofQuantumInformation . .. 22 1.4 InformationisPhysical: TheDilemma . .... 29 1.5 Alternativeapproaches: Dretske . .... 34 1.6 Summary .................................... 39 2 Inadequacy of Shannon Information in QM? 41 2.1 Introduction................................... 41 2.2 Two arguments against the Shannon information . ...... 43 2.2.1 Are pre-existing bit-values required? . ..... 43 2.2.2 Thegroupingaxiom .. .. .. .. .. .. .. .. .. .. .. 47 2.3 Brukner and Zeilinger’s ‘Total information content’ . ........... 54 2.3.1 Some Different Notions of Information Content . .... 56 2.3.2 The Relation between Total Information Content and I(~p)..... 59 2.4 Conclusion ................................... 63 3 Case Study: Teleportation 64 3.1 Introduction................................... 64 3.2 Thequantumteleportationprotocol . .... 65 3.2.1 Some information-theoretic aspects of teleportation......... 67 3.3 Thepuzzlesofteleportation . ... 69 3.4 Resolving(dissolving)theproblem . ..... 71 3.4.1 Thesimulationfallacy . 73 3.5 The teleportation process under different interpretations.......... 76 3.5.1 Collapse interpretations: Dirac/von Neumann, GRW . ...... 77 3.5.2 No collapseandnoextravalues: Everett. ... 78 3.5.3 Nocollapse,butextravalues: Bohm . 80 3.5.4 Ensembleandstatisticalviewpoints . ... 86 i CONTENTS ii 3.6 Concludingremarks .............................. 87 4 The Deutsch-Hayden Approach 92 4.1 Introduction................................... 92 4.2 TheDeutsch-HaydenPicture . 94 4.2.1 Localityclaim(2): Contiguity. .. 99 4.3 AssessingtheClaimstoLocality . .102 4.3.1 TheConservativeInterpretation . .103 4.3.2 TheOntologicalInterpretation . .107 4.4 InformationandInformationFlow . .111 4.4.1 Whereaboutsofinformation. 112 4.4.2 Explaining information flow in teleportation: Locally accessible and inaccessible information114 4.4.3 Assessing the claims for information flow . .117 4.5 Conclusion ...................................123 5 Entanglement in Deutsch-Hayden 126 5.1 Background...................................128 5.1.1 Entanglement witnesses and the Horodecki’s PPT condition . .129 5.1.2 Themajorizationcondition . 134 5.1.3 The tetrahedronof Bell-diagonalstates . .136 5.2 Characterizations
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