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Pure States Statistical Mechanics: on Its Foundations and Applications to Quantum Gravity

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Pure States Statistical Mechanics: on Its Foundations and Applications to Quantum Gravity Pure states statistical mechanics: On its foundations and applications to quantum gravity Fabio Anzà St Catherine’s College University of Oxford arXiv:1808.00535v1 [quant-ph] 1 Aug 2018 A thesis submitted for the degree of Doctor of Philosophy Trinity 2018 I dedicate this work to my grandparents, Nonna Pia and Nonno Cocò. The path you laid down and your endless support gives me the strength to outlast my mistakes. For that, I will always be extraordinarily grateful. Abstract The project concerns the study of the interplay among quantum mechanics, sta- tistical mechanics and thermodynamics, in isolated quantum systems. The goal of this research is to improve our understanding of the concept of thermal equilibrium in quantum systems. First, I investigated the role played by observables and measurements in the emer- gence of thermal behaviour. This led to a new notion of thermal equilibrium which is specific for a given observable, rather than for the whole state of the system. The equilibrium picture that emerges is a generalization of statistical mechanics in which we are not interested in the state of the system but only in the outcome of the mea- surement process. I investigated how this picture relates to one of the most promising approaches for the emergence of thermal behaviour in quantum systems: the Eigen- state Thermalization Hypothesis. Then, I applied the results to study the equilibrium properties of peculiar quantum systems, which are known to escape thermalization: the many-body localised systems. Despite the localization phenomenon, which pre- vents thermalization of subsystems, I was able to show that we can still use the predictions of statistical mechanics to describe the equilibrium of some observables. Moreover, the intuition developed in the process led me to propose an experimentally accessible way to unravel the interacting nature of many-body localised systems. Then, I exploited the “Concentration of Measure” and the related “Typicality Ar- guments” to study the macroscopic properties of the basis states in a tentative theory of quantum gravity: Loop Quantum Gravity. These techniques were previously used to explain why the thermal behaviour in quantum systems is such an ubiquitous phenomenon at the macroscopic scale. I focused on the local properties, their ther- modynamic behaviour and interplay with the semiclassical limit. The ultimate goal of this line of research is to give a quantum description of a black hole which is consis- tent with the expected semiclassical behaviour. This was motivated by the necessity to understand, from a quantum gravity perspective, how and why an horizon exhibits thermal properties. Acknowledgements These three years have been a painfully wonderful journey. I have met amazing people and each one of them has given me something I am thankful for: a few moments of their time. There is absolutely no way I will be able to express, in words, how grateful I am. But I am going to try anyway. I would like to start by thanking the people who have had the patience to work with me. First, my supervisor, Vlatko Vedral, for letting me forge my own re- search path, without interfering, while being a constant source of encour- agement and scientific inspiration. On that note, I am also profoundly grateful to Davide Girolami for always being there when I needed an ad- vice. I will miss, deeply, our chats about physics, your insights and our discussions about Italian politics. It was really fun. I would also like to thank the other members of the Oxford group for providing a wonderful research environment: Tristan Farrow, Oscar Dahlsten, Benjamin Yadin, Christian Schilling, Cormac Browne, the Fe- lixes (or Felices, according to Ben[1]) Binder and Tennie, Nana Liu, Pieter Bogaert, Anu Unnikrishnan, Tian Zhang and Reevu Maity. Many thanks also to my collaborators for helping, teaching and showing me your way: Goffredo Chirco, Christian Gogolin, Marcus Huber and Francesca Pietra- caprina. During some rough times, I was lucky enough to have someone who was willing to give me a piece of advice. For that, I am particularly indebted to John Goold and Michele Campisi. I want you to know your advices gave me the strength to trust my choice, even when others casted doubt on it. Many thanks to all the people and institutions who have hosted me during these three years. In particular, I would like to thank Rosario Fazio and the Condensed Matter Group at ICTP for hosting me in Trieste. A special thank is directed to Jim Crutchfield and the Information Theory group at U.C. Davis: For opening your door, giving me asylum and show- ing me what an amazing environment you created. I am also grateful to the “Angelo Della Riccia” foundation and to St. Catherine’s College for providing funding for my research. Among the amazing people I have met in these years, there is one group which always made me feel at home: the Oxford University Volleyball Club. I will never be able to thank you enough for everything we have experienced together: The game and the adrenaline, the incredibly late nights, the fun on the sand and even the boring line-judging duties during the ladies’ games. We have accomplished much and these marvellous memories will always have a special place in my heart. So, thank you KBar, Alex, Jonas, Gytis, Andy, David, Sanders, Stefan, Rory, Kuba, Christos, Adam, Sven, Nick, Andrea and Megan. I will always hold your friendship in the highest regard. Finally, I would like to thank my parents, my brother and my sister for their endless support. You have given me the strength to make my own mistakes and shown me the wisdom to accept them. I will always love you and I want you to know I could not have done this without you. Last but not least, I would like to thank Carmen for showing up, out of the blue, and for deciding to put up with me every day. It is not easy, but we will have a lot of fun. 5 Contents Preface1 Overview5 1 Introduction 10 1.1 States . 10 1.2 Observables . 11 1.3 Measurements . 12 1.4 Quantum bits . 13 1.5 Time-evolution of isolated quantum systems . 14 1.6 Entropies . 16 1.7 Quantum and classical distinguishability . 17 1.8 Quantum Entanglement . 17 1.9 Quantum thermal equilibrium . 18 2 Pure-states statistical mechanics 20 2.1 Introduction . 20 2.2 Roadmap of recent reviews . 22 2.3 Eigenstate Thermalization . 25 2.3.1 Versions of the ETH . 26 2.3.2 Thermalization according to the ETH . 28 2.3.3 Summary on the ETH . 30 2.4 Typicality . 31 2.4.1 Summary on Typicality . 34 2.5 Summary . 35 i On the foundations 36 3 Information-theoretic equilibrium and Observable Thermalization 37 3.1 Introduction . 37 3.2 Information-theoretic equilibrium . 40 3.3 Relation with statistical mechanics . 42 3.4 Isolated Quantum Systems - Relation to ETH . 43 3.4.1 Hamiltonian Unbiased Observables and ETH . 44 3.4.2 HUOs and ETH: two important examples . 49 3.5 Discussion . 50 4 Eigenstate Thermalization for Degenerate Observables 53 4.1 Physical observables . 54 4.1.1 Hamiltonian Unbiased Observables . 54 4.2 Conclusions . 60 5 Thermal Observables in a Many-Body Localized System 61 5.1 Localised Systems . 61 5.1.1 Anderson Localization . 62 5.1.2 Many-Body Localization . 63 5.1.3 The Q-LIOMs picture . 65 5.2 Thermal observables in MBL . 67 5.2.1 Dynamical study of observable thermalization . 68 5.3 Logarithmic growth of entanglement . 72 5.3.1 Local entropy and total correlations . 72 5.3.2 Logarithmic spread of entanglement . 73 5.3.3 About initial states . 74 5.3.4 Summary . 79 5.4 Summary and Conclusion . 79 Application to Quantum Gravity 81 6 Macroscopic aspects of Spin Networks 82 6.1 Macroscopic behaviour of spin networks . 82 6.2 Spin Network states of quantum geometry . 85 6.2.1 Spin Networks . 85 6.2.2 Quantum Geometry . 88 ii 6.2.3 Area, Volume and fuzzy geometry . 91 7 Typicality in spin-networks 93 7.1 Introduction . 93 7.2 Intertwiner Typicality . 94 7.2.1 Definition of the constraint . 95 7.2.2 The canonical states of the system . 96 7.3 Typicality of the reduced state . 99 7.3.1 Evaluation of the bound . 100 7.3.2 Levy’s lemma . 101 7.4 Thermodynamic limit & area laws . 103 7.5 Summary and Discussion . 106 8 Fate of the Hoop Conjecture 109 8.1 Introduction . 109 8.1.1 A 3D region of quantum space . 110 8.1.2 Separating Boundary and Bulk . 112 8.2 Typicality of the boundary . 113 8.2.1 A simplified setting . 114 8.2.2 Evaluation of the bound . 114 8.2.3 The typical boundary state . 115 8.2.4 Canonical coefficient . 116 8.3 Summary and conclusions . 116 9 Conclusions 119 9.1 Summary . 119 9.2 Conclusions and Future work . 121 A Equilibrium Equations 126 B Levy’s lemma 129 C Proof of Theorem2 130 C.1 Generalised Bloch-vector parametrization . 130 C.2 Proof of Theorem2............................ 132 C.3 Examples . 133 D Intertwiner Hilbert spaces 139 iii Bibliography 142 iv List of Figures 2.1 Setup for the “Local thermalization” paradigm . 32 5.1 Exponentially suppressed support of an l-bit. 66 5.2 Local magnetization in the thermal phase . 70 5.3 Local magnetization in the localized phase . 71 5.4 Time-depedent profile of the rescaled total correlations . 76 5.5 Time-depedent profile of the local entropies . 77 5.6 Bipartite entanglement dynamics in the localised phase .
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