Quantum Information Theory and The Foundations of Quantum Mechanics Christopher Gordon Timpson The Queen’s College arXiv:quant-ph/0412063 v1 8 Dec 2004 A thesis submitted for the degree of Doctor of Philosophy at the University of Oxford Trinity Term 2004 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.com 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).com 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 sûr; 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.com Contents Introduction iii I What is Information? 1 1 Concepts of Information 3 1.1 How to talk about information: Some simple ways .2 The Shannon Information and related concepts .1 Interpretation of the Shannon Information .2 More on communication channels .3 Interlude: Abstract/concrete; technical, everyday .3 Aspects of Quantum Information .4 Information is Physical: The Dilemma .5 Alternative approaches: Dretske. 39 2 Inadequacy of Shannon Information in QM? 41 2.2 Two arguments against the Shannon information .1 Are pre-existing bit-values required? .2 The grouping axiom .3 Brukner and Zeilinger’s ‘Total information content’ .1 Some Different Notions of Information Content .2 The Relation between Total Information Content and I(~ p). 63 3 Case Study: Teleportation 64 3.2 The quantum teleportation protocol .1 Some information-theoretic aspects of teleportation .3 The puzzles of teleportation .4 Resolving (dissolving) the problem .1 The simulation fallacy .5 The teleportation process under different interpretations .1 Collapse interpretations: Dirac/von Neumann, GRW .2 No collapse and no extra values: Everett .3 No collapse, but extra values: Bohm .4 Ensemble and statistical viewpoints .com CONTENTS ii 3.
87 4 The Deutsch-Hayden Approach 92 4.2 The Deutsch-Hayden Picture .1 Locality claim (2): Contiguity .3 Assessing the Claims to Locality .1 The Conservative Interpretation .2 The Ontological Interpretation .4 Information and Information Flow .1 Whereabouts of information .2 Explaining information flow in teleportation: Locally accessible and inaccessible information114 4.3 Assessing the claims for information flow. 123 5 Entanglement in Deutsch-Hayden 126 5.1 Entanglement witnesses and the Horodecki’s PPT condition .2 The majorization condition .3 The tetrahedron of Bell-diagonal states .2 Characterizations in the Deutsch-Hayden representation .1 Some sufficient conditions for entanglement .2 The PPT and reduction criteria. 149 6 Quantum Computation and the C-T Hypothesis 151 6.2 Quantum computation and containing information .3 The Turing Principle versus the Church-Turing Hypothesis .1 Non-Turing computability? The example of Malament-Hogarth spacetimes163 6.4 The Church-Turing Hypothesis as a constraint on physics?. 167 7 Morals 171 II Information and the Foundations of Quantum Mechanics174 8 Preliminaries 176 8.1 Information Talk in Quantum Mechanics.
176 9 Some Information-Theoretic Approaches 183 9.1 Zeilinger’s Foundational Principle .1 Word and world: Semantic ascent .2 Shannon information and the Foundational Principle .2 The Clifton-Bub-Halvorson characterization theorem .2 Some queries regarding the C ∗ -algebraic starting point .3 Questions of Interpretation .com Introduction Much is currently made of the concept of information in physics, following the rapid growth of the fields of quantum information theory and quantum computation. These are new and exciting fields of physics whose interests for those concerned with the foun- dations and conceptual status of quantum mechanics are manifold. On the experimental side, the focus on the ability to manipulate and control individual quantum systems, both for computational and cryptographic purposes, has led not only to detailed re- alisation of many of the gedanken-experiments familiar from foundational discussions (see e. Zeilinger (1999a)), but also to wholly new demonstrations of the oddity of the quantum world (Boschi et al., 1998; Bouwmeester et al., 1997; Furusawa et al.
Developments on the theoretical side are no less important and interesting. Concentra- tion on the possible ways of using the distinctively quantum mechanical properties of systems for the purposes of carrying and processing information has led to considerable deepening of our understanding of quantum theory. The study of the phenomenon of entanglement, for example, has come on in leaps and bounds under the aegis of quantum information (see e. The excitement surrounding these fields is not solely due to the advances in the physics, however.
It is due also to the seductive power of some more overtly philosophical (indeed, controversial) theses. There is a feeling that the advent of quantum information theory heralds a new way of doing physics and supports the view that information should play a more central rôle in our world picture. In its extreme form, the thought is that information is perhaps the fundamental category from which all else flows (a view with obvious affinities to idealism)1 , and that the new task of physics is to discover and 1 Consider, for example, Wheeler’s infamous ‘It from Bit’ proposal, the idea that every physical thing (every ‘it’) derives its existence from the answer to yes-no questions posed by measuring devices: ‘No iii www.com INTRODUCTION iv describe how this information evolves, manifests itself and can be manipulated. Less extravagantly, we have the ubiquitous, but baffling, claim that ‘Information is Physical’ (Landauer, 1996) and the widespread hope that quantum information theory will have something to tell us about the still vexed questions of the interpretation of quantum mechanics.
These claims are ripe for philosophical analysis. To begin with, it seems that the seductiveness of such thoughts appears to stem, at least in part, from a confusion between two senses of the term ‘information’ which must be distinguished: ‘information’ as a technical term which can have a legitimate place in a purely physical language, and the everyday concept of information associated with knowledge, language and meaning, which is completely distinct and about which, I shall suggest, physics has nothing to say.