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Multi-Resource Theories and Applications to Quantum Thermodynamics

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Multi-Resource Theories and Applications to Quantum Thermodynamics Multi-resource theories and applications to quantum thermodynamics By Carlo Sparaciari A thesis submitted to University College London for the degree of Doctor of Philosophy Department of Physics and Astronomy University College London September 17, 2018 2 I, Carlo Sparaciari confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Signed ............................................ Date ................ 3 4 Multi-resource theories and applications to quantum thermodynamics Carlo Sparaciari Doctor of Philosophy of Physics University College London Prof. Jonathan Oppenheim, Supervisor Abstract Resource theories are a set of tools, coming from the field of quantum information theory, that find applications in the study of several physical scenarios. These theories describe the physical world from the perspective of an agent, who acts over a system to modify its quantum state, while having at disposal a limited set of operations. A noticeable example of a physical theory which has recently been described with these tools is quantum thermodynamics, consisting in the study of thermodynamic phenomena at the nano-scale. In the standard approach to resource theories, it is usually the case that the constraints over the set of available operations single out a unique resource. In this thesis, we extend the resource theoretic framework to include situations where multiple resources can be identified, and we apply our findings to the study of quantum thermodynamics, to gain a better understanding of quantities like work and heat in the microscopic regime. We introduce a mathematical framework to study resource theories with multiple resources, and we explore under which conditions these multi-resource theories are reversible. Further- more, we investigate the interconversion of resources, i.e., in which situations it is possible to exchange between two different kinds of resources. We then apply this formalism to quantum thermodynamics, where the two resources under consideration are energy and entropy. This multi-resource theory allows us to explore thermodynamics when the system under examination is closed or coupled with a thermal environment with a finite size. In addition to our work on multi-resource theories, we study the states of equilibrium of closed systems, known as passive states, and analyse under which circumstances it is possible to extract energy from these states. We show that passivity is energetically unstable, and that, even for closed systems, the only stable states are those with a well-defined temperature. 5 6 Impact Statement The results presented in this thesis concern the fields of quantum information theory and quantum thermodynamics. These two branches of physics are both pivotal in the development of quantum technologies, i.e., devices that exploit the quantum features of nature to outperform their classical counterparts. The most renewed representative of these technologies is probably the quantum computer, but other important examples include quantum sensors, microscopic machines, and quantum simulators. Research into these devices has recently started, with giants like Google and IBM working on their own prototypes, with the United Kingdom funding a national network of Quantum Technology Hubs, and with the European Union investing around e1 billion to create a strong community of experts in this area. In order for quantum technologies to become a reality, we need to progress our scientific and engineering knowledge, and in particular we need to fully understand how thermodynamic and information processes work at the quantum scale. My work focuses on a set of tools, known as resource theories, which have been proven to be particularly suitable for describing thermodynamic processes at the microscopic scale and quantum information-processing protocols. With the help of these tools, I have designed ther- modynamic models able to describe physical situations close to those occurring in a laboratory. For example, I considered scenarios where the main system under examination interacts with a surrounding environment which does not have an infinite size, and is not thermal, i.e., is not described by a single parameter, namely, its temperature. These results have the potential to inform new theoretical and experimental work in quantum thermodynamics. Indeed, it should be possible to build upon the ideas introduced in this thesis to develop new non-idealised models to describe thermodynamics of physical systems, for instance where correlations with the envi- 7 ronment are considered. Furthermore, some of the protocols developed in the thesis could be realised in the laboratory, for example to increase the efficiency of energy-extracting machines. Another part of my work concerns the development of a theoretical framework able to describe tasks where several quantities, or resources, are involved. This work widens the scope of applicability of resource theories, and introduces new concepts { such as the notions of banks and resource interconversion { which could help the theoretical research in this field. Furthermore, this framework provides tools that could be used to gain new quantitative insights into the properties of many-body systems, and in the fundamental resources responsible for the computational quantum advantage. 8 List of Publications and Preprints The work presented in this thesis contains material from the following publications and pre- prints: 1. C. Sparaciari, J. Oppenheim, and T. Fritz, \Resource theory for work and heat," Physical Review A, vol. 96, no. 5, p. 052112, 2017. 2. C. Sparaciari, D. Jennings, and J. Oppenheim, \Energetic instability of passive states in thermodynamics," Nature Communications, vol. 8, no. 1, p. 1895, 2017. 3. C. Sparaciari, L. del Rio, C. M. Scandolo, P. Faist, and J. Oppenheim, \The first law of general quantum resource theories", arXiv:1806.04937 [quant-ph], 2018. Other publications by the author are: 4. M. Horodecki, J. Oppenheim, and C. Sparaciari, \Extremal distributions under approxi- mate majorization", Journal of Physics A: Mathematical and Theoretical, vol. 51, no. 30, p. 305301, 2018. 9 10 Acknowledgments These three years of Ph.D. have been both an exciting and challenging time of my life, and concluding this path would have been impossible without the professional and emotional support I received from many people. First and foremost, I would like to thank Jonathan, my supervisor, for the many chats about physics we had, and for the freedom he gave me in following my interests and doing my research, while always being available for discussion and guidance. I am grateful to everyone in my group, for creating a great environment to do research and learn new things. In particular, I want to thank Alvaro´ for the infinite discussions on thermodynamics and resource theories, and for being the best travel companion I could have hoped for, for almost every conference I went to. I am also grateful to all the collaborators I had during these years, from whom I have learnt a lot. Particularly, I want to thank L´ıdia,who is one of the best researchers I have ever met, from both a scientific and human point of view. I am also extremely grateful to everyone involved in the CDT, and in particular to Andrew, Dan, and Lopa, for sharing with us their research and passion. I thank my examiners, Alessio Serafini and Giulio Chiribella, for their comments, which considerably improved this thesis. I want to thank all the first cohort of the CDT, for sharing the joy and pain of the first year; from the travel to Whistler to the business case study. The strongest bonds I have in the UK are with some of these people; thanks to Andrew, Cameron, Claudia, Johnnie, Josh, Lorenzo, Sherif, Tim, and Tom. Likewise, I am grateful to Alex, Mike, and Tom, who made working in an overcrowded office a positive and fun experience. Finally, I would like to thank my family for their constant love and support; without it, I would not have achieved much in life. And my deepest thank you is for Paola. You shared with me the best moments of my life, and your love has helped me overcome the worst ones. 11 12 Contents Introduction 21 Structure of the thesis.................................... 24 Basic definitions and notation................................ 27 I Background 31 1 The resource theoretic approach 33 1.1 The structure of a resource theory.......................... 36 1.2 The Second Law of resource theories: monotones.................. 39 1.3 Resource theories in the many-copy regime..................... 41 1.4 Reversibility and resource theories.......................... 47 1.5 Single-copy regime and catalytic transformations.................. 53 1.5.1 Necessary and sufficient conditions for state transformations........ 53 1.5.2 Catalytic transformations........................... 56 2 Thermodynamics as the resource theory of athermality 59 2.1 Thermodynamic setup and Thermal Operations................... 63 2.2 Physical features of Thermal Operations....................... 67 2.2.1 No correlations between system and environment.............. 67 2.2.2 High degree of control over the environment................. 68 2.2.3 Exact energy conservation........................... 69 2.2.4 Controlling the system's Hamiltonian.................... 71 2.2.5 Creating coherence............................... 72 13 2.2.6 Thermalisation and the free states...................... 76 2.3 Thermodynamic monotones.............................
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