Henrik Wilming A Quantum of Thermodynamics From ground state cooling to spontaneous symmetry breaking Im Fachbereich Physik der Freien Universität Berlin eingereichte Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften. Berlin, 2017 Erstgutachter: Prof. Dr. Jens Eisert Zweitgutachter: Prof. Felix von Oppen, PhD Datum der Disputation: 19.7.2018 Publications of the author of this thesis Peer-reviewed publications: [1] H. Wilming, R. Gallego, and J. Eisert. “Second law of thermodynamics under con- trol restrictions”. Phys. Rev. E 93.4 (2016). DOI: 10.1103/physreve.93. 042126. [2] R. Gallego, J. Eisert, and H. Wilming. “Thermodynamic work from operational principles”. New J. Phys. 18.10 (2016), p. 103017. DOI: 10 . 1088 / 1367 - 2630/18/10/103017. [3] M. Perarnau-Llobet, A. Riera, R. Gallego, H. Wilming, and J. Eisert. “Work and entropy production in generalised Gibbs ensembles”. New J. Phys. 18.12 (2016), p. 123035. DOI: 10.1088/1367-2630/aa4fa6. [4] H. Wilming, M. J. Kastoryano, A. H. Werner, and J. Eisert. “Emergence of spon- taneous symmetry breaking in dissipative lattice systems”. J. Math. Phys. 58.3 (2017), p. 033302. DOI: 10.1063/1.4978328. [5] F. Pastawski, J. Eisert, and H. Wilming. “Towards Holography via Quantum Source- Channel Codes”. Phys. Rev. Lett. 119.2 (2017), p. 020501. DOI: 10 . 1103 / PhysRevLett.119.020501. [6] J Lekscha, H Wilming, J Eisert, and R Gallego. “Quantum thermodynamics with local control”. Phys. Rev. E 97.2 (2018), p. 022142. DOI: 10.1103/PhysRevE. 97.022142. arXiv: 10.1103/PhysRevD.97.086015. [7] H. Wilming, R. Gallego, and J. Eisert. “Axiomatic Characterization of the Quan- tum Relative Entropy and Free Energy”. Entropy 19.6 (2017), p. 241. DOI: 10. 3390/e19060241. [8] H. Wilming and R. Gallego. “Third Law of Thermodynamics as a Single Inequal- ity”. Phys. Rev. X 7 (4 2017), p. 041033. DOI: 10 . 1103 / PhysRevX . 7 . 041033. [9] M. Perarnau-Llobet, H. Wilming, A. Riera, R. Gallego, and J. Eisert. “Strong Cou- pling Corrections in Quantum Thermodynamics”. Phys. Rev. Lett. 120.12 (2018). DOI: 10.1103/physrevlett.120.120602. arXiv: 1704.05864. [11] P. Boes, H. Wilming, J. Eisert, and R. Gallego. “Statistical ensembles without typ- icality”. Nature Comm. 9.1 (2018), p. 1022. DOI: 10.1038/s41467- 018- 03230-y. arXiv: 1707.08218. [12] R. Gallego, H. Wilming, J. Eisert, and C. Gogolin. “What it takes to avoid equili- bration”. Phys. Rev. A 98.2 (2018). DOI: 10.1103/physreva.98.022135. arXiv: 1711.09832. Pre-prints: [10] H. Wilming, M. Goihl, C. Krumnow, and J. Eisert. “Towards local equilibration in closed interacting quantum many-body systems” (2017). arXiv: 1704.06291 [quant-ph]. Contents 1 Introduction and Overview 11 2 Thermodynamics as a resource theory 17 2.1 Formalizing the model 18 2.2 Example: Anomalous heat-flow 19 2.3 Monotones, the data-processing inequality and the second laws 22 2.4 Illustration: Landauer Erasure and side-information 24 2.5 Illustration: The infinite ladder battery and fluctuations of energy 26 2.6 Connection to thermodynamics in terms of open system dynamics 28 2.7 Conclusions and Outlook 29 3 Ground state cooling and the unattainability principle 31 3.1 Definition and basic properties of the vacancy 33 3.2 Sufficiency: A general theorem 34 3.3 i.i.d. resources and the scaling of the vacancy 35 3.4 Thermal resources 37 3.5 Exact and approximate catalysts 38 3.6 Coming back to Landauer erasure and energy measurements 39 3.7 Conclusion and outlook 41 4 What is work? 43 4.1 Deterministic changes of energy as heat 44 4.2 Thermodynamic tasks and work-storage devices 45 4.3 An operational approach 45 4.4 The Axioms 48 4.5 Functions that fulfill the Axioms 49 4.6 Irreversibility and the second law 50 4.7 The role of correlations and super-additivity 51 4.8 Probabilistic work 53 4.9 Summary 54 5 Uniqueness of the relative entropy and free energy 55 5.1 Gibbs-preserving maps 56 5.2 Catalysts and Correlations 58 8 CONTENTS 5.3 Monotones and the free energy 59 5.4 Proof of theorem 5.1 61 5.5 Conclusions and outlook 62 6 Statistical ensembles from thermodynamic operations 63 6.1 Macrostates and macrostate operations 64 6.2 Microstate operations and operational equivalence 66 6.3 Example application: Work extraction bounds 68 6.4 Example application: Deriving the Clausius inequality 69 6.5 Conclusions and relation to Jaynes’ principle and typicality results 69 7 Equilibration in many-body systems 71 7.1 Brief review of equilibration and thermalization in closed quantum systems 71 7.2 Equilibration as dephasing and the problem of timescales 75 7.3 The basic mechanism 76 7.4 Refining the notion of equilibration 78 7.5 Connecting the discussion to general results 81 7.6 Conclusions and Outlook 86 8 The cost of avoiding equilibration 87 8.1 Formalization of the problem and main result 88 8.2 Non-stationary initial states 90 8.3 Correlations and the second law of equilibration 91 8.4 Impossibility equilibration bounds in terms of von Neumann entropy 95 8.5 Conclusions and Outlook 95 9 Leaving the ideal world: Thermal machines under control restrictions 97 9.1 Simplifying the model 98 9.2 The fundamental bound for a single heat bath 101 9.3 Thermalizing contacts are not universal 102 9.4 Thermal machines and efficiencies 105 9.5 Local control: Implications from the theory of quantum control 107 9.6 Illustration: The classical one-dimensional Ising model 108 9.7 Conclusions and Outlook 113 10 Thermodynamics in closed quantum systems using Generalized Gibbs En- sembles 115 10.1 Generalized Gibbs Ensembles as equilibration models 115 10.2 Thermodynamic protocols 117 10.3 Entropy production and reversibility 120 10.4 The minimum work principle 126 10.5 Summary 133 AQUANTUMOFTHERMODYNAMICS 9 11 Corrections to work and efficiency under strong coupling 135 11.1 Work extraction in the strong coupling regime 136 11.2 Efficiency: Thermal machines in the strong coupling regime 144 11.3 Power 147 11.4 Example: Quantum Brownian Motion 148 11.5 Summary 150 12 On spontaneous symmetry breaking in dissipative systems 151 12.1 Formal set-up 152 12.2 Discrete symmetries 155 12.3 Continuous symmetries 157 12.4 Goldstone-modes 159 12.5 Time-scales 160 12.6 Conclusions 161 13 Conclusions and Acknowledgments 163 14 Technical proofs und supplementary information 165 14.1 Thermodynamics as a resource theory 165 14.2 Ground state cooling and the unattainability principle 167 14.3 What is work? 173 14.4 Uniqueness of the relative entropy and free energy 179 14.5 Statistical ensembles from thermodynamic operations 180 14.6 Thermal machines under control restrictions 183 14.7 Thermodynamics in closed quantum systems using GGEs 189 14.8 Corrections to work and efficiency under strong coupling 190 14.9 On spontaneous symmetry breaking in dissipative systems 192 15 Bibliography 199 A Back matter 213 A.1 Summary of the thesis 213 A.2 Zusammenfassung 214 A.3 Anteil des Autors bei Konzeption, Durchführung und Verfassung der zugrunde- liegenden Arbeiten 215 A.4 Eigenständigkeitserklärung 216 1 Introduction and Overview THERMODYNAMICS is a tremendously useful theory. It can be used to explain the func- tioning of steam engines, refrigerators, power plants and cars. But at the same time it also explains the very basic mechanism behind climate-change (together with some basic prop- erties of the interaction of certain molecules with light), chemical reactions, or why it helps to put ice-cubes in water if we want to have a cool drink. But it does not stop there: In fact, today it is also believed that even properties of black holes can and indeed have to be understood thermodynamically [13–15] to get a consistent physical description of our universe. Indeed, this is by now one of the guiding principles for obtaining a viable the- ory of quantum gravity and was essential for the currently much-discussed "Holographic" approach to quantum gravity [16–20]. The success and universality of thermodynamics is surprising given the fact that it was invented in the 19th century – long before quantum mechanics and statistical physics were developed. This was possible since the concepts and principles of thermodynamics can in fact be developed without any reference to the underlying microscopic physical mech- anisms that give rise to them: It is essentially a formal framework based on very general axioms [21–24]. Moreover, these axioms or principles, often called the laws of thermo- dynamics, can be stated in an operational way. This means that they state that certain operations or processes on physical objects are impossible to perform (creating energy out of nothing, lifting arbitrary weights only using a single heat bath, cooling a system to zero temperature with finite resources). All predictions of phenomenological thermodynamics can be derived from these basic principles together with experimentally measurable quan- tities describing matter at the level of thermodynamic quantities (the equations of state). Of course, today we know that it is quantum mechanics and not thermodynamics that describes the material world around us on the fundamental level. Once we accept that quantum mechanics underlies our physical reality at the microscopic level, we must then also accept that the thermodynamic principles have to follow from quantum mechanics, even though they can be formulated without referring to it. From a theoretical point of view it is then most desirable to understand exactly how thermodynamics emerges from quantum mechanics. In particular, the following questions have to be answered: 1. What exactly is (thermal) equilibrium and how does it emerge? 2.