Emergence of a classical world from within quantum theory by David Poulin A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2004 c David Poulin 2004 Author’s declaration I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or indi- viduals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. iii Thesis Committee This thesis has been evaluated, approved without modifications, and nominated for the Pearson medal on November 26th of the year 2004 by a jury composed of the following supervisors and examiners: • William B. Anderson, Examination Chair Department of Chemical Engineering, University of Waterloo • Artur Ekert, External Examiner Department of Applied Mathematics & Theoretical Physics, Cambridge University • Raymond Laflamme, Supervisor Department of Physics, University of Waterloo • Robert Mann, Member of Committee Department of Physics, University of Waterloo • Michele Mosca, Internal-External Examiner Combinatorics & Optimization Department, University of Waterloo • Lee Smolin, Member of Committee Department of Physics, University of Waterloo v Abstract The starting point of this dissertation is that a quantum state represents the ob- server’s knowledge about the system of interest. As it has been pointed out several times by the opponents of this epistemic interpretation, it is difficult to reconcile this point of view with our common notion of “physical reality”, which exists independently of our monitoring, and can be discovered without disturbance. Indeed, if quantum theory is correct, it should apply to classical systems — including measurement devices — as well as to any other system. In this dissertation, we will study the quantum mechanisms responsible for our per- ception of the world and demonstrate how they lead to the emergence of an operational objective reality from within quantum theory: several observers gathering information through these mechanisms will arrive at a common consensus about the properties of the world. The two mechanisms we study in great detail are the redundant proliferation of in- formation in the environment and the direct measurement of a macroscopic observable. An example of the first mechanism is the photon environment which provides us with our visual data about the world. Several independent observers learning about their surroundings in this indirect fashion will agree on their findings. An example of the sec- ond mechanism is our tactile information: when the tip of our finger touches an object, it interacts collectively with a very large number of molecules. Again, under realistic assumptions, this type of information acquisition will lead to a classical perception of the world. Key words: Quantum-classical transition, Interpretation of quantum theory, Decoher- ence, Macroscopic observables, Weak measurements, Ensemble measurements, Nuclear magnetic resonance quantum information processing, Quantum information. vii R´esum´e Le point de d´epart de la pr´esente dissertation est qu’un ´etat quantique repr´esente la connaissance d’un observateur par rapport au syst`eme `al’´etude. Comme l’ont fait valloir les principaux opposants `acette interpr´etation dite ´epist´emique de la th´eorie quantique, il est difficile de r´econcilier ce point de vue avec notre notion commune de r´ealit´ephysique, r´ealit´equi existe de fa¸con ind´ependante de l’observation et qui peut ˆetred´ecouverte sans ˆetreperturb´ee. En effet, si nous admettons que la th´eorie quantique est correcte, alors elle doit convenir `atous les syst`emes physiques, aussi bien classiques que quantiques, dont en particulier les appareils de mesure. Dans cette dissertation, nous pr´esentons une ´etude d´etaill´eedes principaux m´ecanis- mes quantiques `al’aide desquels nous percevons notre univers. Nous d´emontrons com- ment une r´ealit´eobjective peut ´emerger de ces m´ecanismes de fa¸conop´erationnelle: divers observateurs ind´ependants recueillant de l’information `al’aide de ces m´ecanismes arriverons `aun consensus commun sur les propri´et´es de notre univers. Nous identifions deux types de m´ecanismes responsables pour notre perception du monde: l’acquisition indirecte d’information par le billais de la prolif´erationd’information dans l’environnement et la mesure directe d’une quantit´emacroscopique. A` titre d’exem- ple du premier m´ecanisme, citons l’environnement de photons responsable de toutes nos donn´ees visuelles. Plusieurs observateurs ind´ependants recueillant cette information visuelle arrivent toujours a un consensus sur les propri´et´esdu monde qui les entour. Le second m´ecanisme peut d´ecrire par exemple toute l’information tactile que nous recevons: lorsque la pointe de notre doigt touche un objet, elle interagit de fa¸concol- lective avec un tr`esgrand nombre de mol´ecules. Encore une fois, ce type d’acquisition d’information nous conduit `aune perception classique de notre univers. Mots cl´es: Transition quantique-classique, Interpretation de la th´eorie quantique, D´eco- h´erence, Observables macroscopiques, Mesures faibles, Mesures d’ensemble, Manipula- tion d’information quantique par r´esonance magn´etique nucl´eaire,Information quan- tique. viii Contents 1 Motivation 1 1.0.1 The epistemic view . 2 1.0.2 Distinction between quantum and classical . 3 1.0.3 Two questions . 5 1.0.4 Our proposal . 6 1.1 Outline of the dissertation . 8 1.2 Additional information for the examiners of this thesis . 9 2 Environment as a witness 13 2.1 Introduction . 13 2.2 Objectivity: what is the problem? . 15 2.3 Definitions and conventions . 19 2.4 Information . 20 2.4.1 Correlations between system and environment . 22 2.4.2 Redundancy of information in the environment . 24 2.4.3 Fragments of the environment and elementary subsystems . 24 2.5 Consequences of redundancy . 26 2.6 Quantum Darwinism: Dynamical emergence of objectivity . 33 2.6.1 Dynamics of Quantum Darwinism and decoherence . 34 2.6.2 Decoherence: the focus on the system . 36 2.6.3 Perfect correlations . 36 2.6.4 Imperfect correlations . 40 2.6.5 Quantum Darwinism exemplified . 42 2.7 Discussion . 45 2.7.1 Assumptions . 46 2.7.2 The state ρSE ............................. 51 2.8 Summary and Conclusion . 54 ix 3 Macroscopic observables 57 3.1 Overview . 57 3.2 Definitions . 59 3.2.1 Method of types . 59 3.2.2 Macroscopic observable . 61 3.2.3 Coarse grained macroscopic POVMs . 64 3.3 Type measurement on identically prepared systems . 66 3.3.1 State disturbance . 67 3.3.2 Gaussian smoothing . 71 3.3.3 General smoothing . 72 3.3.4 Mixed states . 75 3.4 Exchangeability . 75 3.4.1 Bulk tomography . 77 3.5 Classicality . 79 3.6 NMR information processing . 85 3.7 Conclusion and open questions . 89 4 Conclusions 93 A Environment as a witness: mathematical complement 97 B Macroscopic observables: mathematical complement 101 B.1 Single molecule post-measurement state . 101 B.2 Conditional fidelity . 103 C Other work published during Ph.D. program 107 C.1 Robust polarization-based quantum key distribution over collective-noise channel† .................................... 107 C.2 Exponential speed-up with a single bit of quantum information: Measur- ing the average fidelity decay† ........................ 117 C.2.1 Efficient algorithm . 121 C.2.2 Entanglement . 122 C.2.3 Quantum probe . 123 C.2.4 Numerical study . 124 x C.2.5 Conclusion . 124 C.3 Estimation of the Local Density of States on a Quantum Computer† .. 127 C.4 Testing integrability with a single bit of quantum information† ..... 137 C.4.1 Overview . 138 C.4.2 Level distribution . 139 C.4.3 Quantum algorithm . 143 C.4.4 Numerical results . 144 C.4.5 Conclusion . 149 C.5 Compatibility of quantum states† ...................... 151 C.5.1 Measurements . 157 C.5.2 Combining knowledge . 158 C.5.3 Knowledge . 159 C.6 Noiseless Subsystems for Collective Rotation Channels in Quantum In- formation Theory† .............................. 161 C.6.1 Introduction . 162 C.6.2 Background . 163 C.6.3 Collective Rotation Channels . 166 C.6.4 Commutant Structure Theorem . 169 C.6.5 Generalized Collective Rotation Channels . 177 xi List of Figures 2.1 Venn diagram for classical information. 23 2.2 Quantum Darwinism exemplified. 43 3.1 State disturbance for general smoothing. 74 C.1 Quantum circuit for the fidelity decay . 122 C.2 Fidelity decay for the kicked top . 125 C.3 Quantum circuit for the LDOS . 131 C.4 Quantum circuit for evaluating the trace . 143 C.5 Form factor for random matrices . 145 C.6 Kicked top in regular regime . 147 C.7 Kicked top in chaotic regime . 147 C.8 Graphical representation of T r{Fˆ} ..................... 148 C.9 Transition from regular to chaotic . 148 C.10 Structure of the commutant . 170 xiii Acknowledgments Raymond Laflamme has been for the past three years a great research supervisor, a pretty good mountain bike partner, and a dear friend. Through the dynamical research environment he has built, by supporting my many travels, and by continually expressing his confidence in me, he has helped me to establish new and exciting research projects as well as to maintain old ones well alive. But most importantly, he always has the perfect comment or just the right question to ask that adds an extra twist to any research program. Working with him has been greatly stimulating and remarkably enriching. Many thanks to Harold Ollivier and Wojciech Zurek, my two partners in crime, with who a part of this dissertation has been elaborated. The least I can say is this has been a very entertaining collaboration, through which I have learned enormously.