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Phd Thesis, University of Cambridge, 2007 Quantum Causal Structure and Quantum Thermodynamics By MIRJAM SARAH WEILENMANN DOCTOR OF PHILOSOPHY UNIVERSITY OF YORK MATHEMATICS OCTOBER 2017 ABSTRACT his thesis reports progress in two domains, namely causal structures and microscopic thermodynamics, both of which are highly pertinent in the development of quantum Ttechnologies. Causal structures fundamentally influence the development of protocols for quantum cryptography and microscopic thermodynamics is crucial for the design of quantum computers. The first part is dedicated to the analysis of causal structure, which encodes the relationship between observed variables, in general restricting the set of possible correlations between them. Our considerations rely on a recent entropy vector method, which we first review. We then develop new techniques for deriving entropic constraints to differentiate between causal structures. We provide sufficient conditions for entropy vectors to be realisable within a causal structure and derive new, improved necessary conditions in terms of so-called non-Shannon inequalities. We also report that for a family of causal structures, including the bipartite Bell scenario and the bilocal causal structure, entropy vectors are unable to distinguish between classical and quantum causes, in spite of the existence of quantum correlations that are not classically reproducible. Hence, further development is needed in order to understand cause from a quantum perspective. In the second part we explore an axiomatic framework for modelling error-tolerant processes in microscopic thermodynamics. Our axiomatisation allows for the accommodation of finite precision levels, which is crucial for describing experiments in the microscopic regime. Moreover, it is general enough to permit the consideration of different error types. The framework leads to the emergence of manageable quantities that give insights into the feasibility and expenditure of processes, which for adiabatic processes are shown to be smooth entropy measures. Our framework also leads to thermodynamic behaviour at the macroscopic scale, meaning that for thermodynamic equilibrium states a unique function provides necessary and sufficient conditions for state transformations, like in the traditional second law. 3 LIST OF CONTENTS Page Abstract 3 List of Contents 5 List of Tables 9 List of Figures 11 Acknowledgements 11 Author’s Declaration 13 1 Introduction and Synopsis 15 2 Preliminaries 21 2.1 Convex geometry and optimisation............................. 22 2.1.1 Convex and polyhedral cones............................ 22 2.1.2 Projections of convex cones: Fourier-Motzkin elimination........... 23 2.1.3 Convex optimisation................................. 27 2.2 Information-theoretic entropy measures ......................... 28 2.2.1 Shannon and von Neumann entropy ....................... 29 2.2.2 Rényi entropies.................................... 30 2.2.3 Smooth entropy measures ............................. 31 2.3 Causal structures....................................... 34 2.3.1 Classical causal structures as Bayesian networks............... 35 2.3.2 Causal structures involving quantum and more general resources . 39 2.4 Quantum resource theories for thermodynamics..................... 42 2.4.1 The resource theory of adiabatic processes.................... 44 5 LIST OF CONTENTS I Entropic analysis of causal structures 47 3 Outline of the entropic techniques for analysing causal structures 49 3.1 Motivation for the entropic analysis............................ 49 3.2 Entropy vector approach................................... 50 3.2.1 Entropy cones without causal restrictions.................... 50 3.2.2 Entropy cones for causal structures........................ 54 3.3 Entropy vector approach with post-selection....................... 64 3.3.1 Post-selection in classical causal structures................... 65 3.3.2 Post-selection in quantum and general non-signalling causal structures . 70 3.4 Alternative techniques for analysing causal structures................. 72 3.4.1 Entropy cones relying on Rényi and related entropies............. 72 3.4.2 Polynomial inequalities for compatible distributions.............. 73 3.4.3 Graph inflations ................................... 73 3.4.4 Semidefinite tests relying on covariance matrices ............... 74 3.5 Appendix ............................................ 74 3.5.1 Inequalities from Example 3.2.14......................... 74 3.5.2 Entropy inequalities and extremal rays for Example 3.3.7.......... 76 4 Inner approximations to the entropy cones of causal structures 79 4.1 Techniques to find inner approximations for causal structures with up to five observed variables....................................... 80 4.2 Inner approximation to the entropy cone of the classical triangle scenario . 82 4.2.1 Inner approximation by means of linear rank inequalities.......... 82 4.2.2 Improving inner approximations ......................... 84 4.3 Appendix ............................................ 85 4.3.1 Inner approximations to the marginal cones of the causal structures of Figure 4.1 ....................................... 85 4.3.2 Technical details regarding Example 4.1.2.................... 88 4.3.3 Proof of Proposition 4.2.1.............................. 90 4.3.4 Inequalities defining the inner approximations in Example 4.2.2 . 91 5 Exploring the gap between inner and outer approximations with non-Shannon inequalities 93 5.1 Improving outer approximations with non-Shannon inequalities........... 93 5.2 Improved entropic approximation for the classical triangle scenario......... 95 5.2.1 Motivating the search for improved outer approximations to the marginal cone of the triangle causal structure ....................... 95 6 LIST OF CONTENTS 5.2.2 Improved outer approximations to the entropy cone of the triangle causal structure........................................ 96 5.3 Non-Shannon inequalities in quantum and hybrid causal structures . 100 5.3.1 Quantum triangle scenario.............................100 5.3.2 Hybrid triangle scenarios..............................104 5.4 Non-Shannon inequalities and post-selection ......................106 5.5 Appendix ............................................107 5.5.1 Proof of Proposition 5.2.5..............................107 5.5.2 Proof of Proposition 5.2.6..............................108 5.5.3 Proof of Lemma 5.3.2 ................................108 6 Entropic distinction between classical and quantum causal structures 111 6.1 Inability of the entropy vector approach to distinguish classical and quantum causes in the bilocal causal structure ...........................111 6.2 Inability of the entropy vector method to distinguish classical and quantum causes in line-like causal structures ................................115 6.3 Towards a general statement ................................116 6.4 Distinguishing classical and quantum causes with post-selection . 118 6.5 Appendix ............................................119 6.5.1 Proof of Theorem 6.2.1................................119 6.5.2 Proof of Proposition 6.3.1..............................123 II Error-tolerant approach to microscopic thermodynamics 125 7 Axiomatic framework for error-tolerant resource theories 127 7.1 Motivation ...........................................127 7.2 Thermodynamic processes with error-tolerance .....................128 7.2.1 The noise-tolerant resource theory of adiabatic processes . 129 7.3 Quantifying resources ....................................132 7.3.1 Smooth entropies for the resource theory of smooth adiabatic operations . 137 7.4 Alternative types of errors in adiabatic operations ...................138 7.5 Appendix ............................................141 7.5.1 Proof of Proposition 7.2.5..............................141 7.5.2 Proof of Proposition 7.3.10 .............................143 7.5.3 Proof of Lemma 7.3.11................................144 7.5.4 Properties of smooth entropies measured with meter systems that achieve the work-error trade-off...............................144 7.5.5 Proof of Lemma 7.3.12................................146 7.5.6 Proof of Proposition 7.3.13 .............................147 7 LIST OF CONTENTS 7.5.7 Proof of Lemma 7.3.15................................149 8 Macroscopic thermodynamics from microscopic considerations 151 8.1 States with macroscopic behaviour.............................151 8.2 Thermodynamic equilibrium states under adiabatic processes . 154 8.3 Appendix ............................................156 8.3.1 Proof of Proposition 8.1.3 and Corollary 8.1.4..................156 8.3.2 Proof of Proposition 8.1.5..............................158 9 Conclusions and Outlook 161 Bibliography 165 8 LIST OF TABLES TABLE Page 2.1 Compatible distributions for three variable causal structures............... 39 9 LIST OF FIGURES FIGURE Page 2.1 Projections of a polyhedral cone................................. 26 2.2 Illustration of parameters introduced to prove Lemma 2.2.8................ 33 2.3 Pearl’s instrumental scenario .................................. 35 2.4 Causal structures with three observed variables....................... 35 2.5 Insufficiency of conditional independencies for characterising causal links . 37 3.1 Illustration of the instrumental causal structure, the bipartite Bell causal structure and the triangle causal structure................................ 55 3.2 Post-selected instrumental and post-selected Bell causal structure............ 66 3.3 Variations
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