Research Collection Doctoral Thesis Restricted agents in thermodynamics and quantum information theory Author(s): Krämer Gabriel, Lea Philomena Publication Date: 2016 Permanent Link: https://doi.org/10.3929/ethz-a-010858172 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 23972 Restricted agents in thermodynamics and quantum information theory A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by Lea Philomena Kr¨amer Gabriel MPhysPhil, University of Oxford born on 18th July 1990 citizen of Germany accepted on the recommendation of Renato Renner, examiner Giulio Chiribella, co-examiner Jakob Yngvason, co-examiner 2016 To my family Acknowledgements First and foremost, I would like to thank my thesis supervisor, Prof. Renato Renner, for placing his trust in me from the beginning, and giving me the opportunity to work in his group. I am grateful for his continuous support and guidance, and I have always benefited greatly from the discussions we had | Renato without doubt has a clear vision, a powerful intuition, and a deep understanding of physics and information theory. Perhaps even more importantly, he has an exceptional gift for explaining complex subjects in a simple and understandable way. I would also like to thank my co-examiners Giulio Chiribella and Jakob Yngvason for agreeing to be part of my thesis committee, and for their input and critical questions in the discussions and conversations we had. Here in Zurich, I primarily want to thank the members of the Quantum Information Group, and the whole of the ITP, for the many enjoyable moments we shared in the last five years; from coffee breaks to intensive exam marking sessions, I have always felt part of a great community that held together and watched out for each other, and I am sure that the bonds and friendships we formed will last for a long time to come. In particular, I am very grateful for the “coffee office” of Caterina, Gizem, and Simone, for the cakes by Aleks and with Reimar, Sandra, Luca, Paolo and others, for the greek coffees with Philippe, and for the many things I learnt during teaching assistance with and from Babis, Manfred and Renato. I want to mention also earlier members of the group, such as Marco, Fr´ed,Cyril, Roger, Johan, Fernando, Norm and Mario Ziman, and members of the Computational Physics group like Tama and Andreas, who have contributed greatly to the welcoming atmosphere I felt at the beginning of my PhD. I would like to give special thanks to Philippe and L´ıdia,my office mates during most of my PhD, who have worked with me on the projects related to this thesis. I have learnt a lot from them, and always enjoyed discussing research | this also applies to Mirjam, who it was always a great pleasure to work with, and to other members of our group like Joe, Sandra, Rotem, Daniela, Christopher and David. Likewise, I have benefited greatly from discussions with members of the wider Quantum Information community at a number of conferences, in particular Jens Eisert, David Jennings, Tobias Fritz, Barry Sanders, Bob Coecke, Dominik Janzing, Ralph, Yelena, Jonathan and Priyaa. Thanks also to L´ıdia,Gilles and Philipp for their feedback on earlier drafts of this thesis, and for the many discussions throughout my PhD. I would also like to thank my new colleagues at JB, who have made me feel extremely welcome at my new job during the critical overlapping first, or last, month of work. Looking back at the past five years, I am also immensely grateful to my friends both at ETH and outside, who have filled my life in Zurich with joy and supported me in difficult times. To this end, I would like to thank my close friend and first Zurich flatmate Ellen, who already studied with me in Oxford, and who really gave me the idea of coming to Zurich. I would also like to thank Lena, my other flatmate and almost-colleague, for her friendship and the many ice creams we had together on sunny days on Hoenggerberg. I am also immensely grateful to my \Br¨uderchen", v friend, dance partner and flatmate Vassili, to my incredible and beloved all-round friend L´ıdia,and to my friends Yasmine, Gilles, Alexejs, crazy Rinat, Manos Cop- perfield (my wine expert who can cook even without onions), who have really made my time in Zurich. The same is true for the Zurich Tango community, and in partic- ular the people of Cuartito Azul, such as Angela, Philip, Alexey, Jorge, Diego and Soledad, Tomas and the Cafet´ınde Buenos Aires, Pablo from El Social, Marcello, Chiara and the other Mates, Olga, Markus, Javier & Rahel, Silvana, Mozzarella and Andrea, Alessandro, Jackie and many more, as well as the Flamenco cajon group. I would also like to thank my friends from before Zurich, such as Lam, Neven, Karl, Hans, Alex & Martha, and in particular Fanny, who have all continued to take part in my life here. Thanks also to the welcoming staff and owners of Caf´eHoengg and Caf´eMiyuko, where parts of this thesis were written. Most importantly, I would like to thank my family, both old and new, in Germany, Hong Kong and Brazil, to whom this thesis is really dedicated | every part of this thesis belongs to you, and would not have been possible without you. It is an incredible blessing to have you all | Nissar, Mama, Papa, Till, IC, Oma, Opa, Kitty, Gotti, Gotti Gila, Gisela, Gottfried, Klaus, Uschi, Mae, Thais, Rodrigo, Martha, Daniel, Marina, Alexia, In´acio,e todo o resto da nossa grande familia! vi Abstract A simple yet powerful way to address most questions in physics and information theory is to ask: \What is possible, and what is impossible, for an agent operating under a given set of constraints? And what would become possible if the agent had access to an additional set of resources?" This idea is at the heart of resource theo- ries, a tool that was first introduced in quantum information theory in the context of entanglement theory, where it gave rise to the well-known concepts of entanglement of formation and distillation, and has found many applications since. For example, thermodynamics has long been understood as a theory that imposes fundamental constraints, like the second law of thermodynamics. Generalising this idea, resource theories of quantum thermodynamics have now shown that microscopic processes are governed by a whole family of second laws, which involve single-shot entropies known from information theory. While resource theories have greatly advanced our understanding of many areas in physics, there are a few important aspects that have not yet been addressed explicitly within existing resource theories; it is the main aim and contribution of this thesis to fill these gaps. The first aspect concerns the subjective knowledge of agents. In quan- tum information theory, this is normally represented by means of density matrices | this does not admit approximate descriptions, coarse-grained descriptions such as specifying a few observable properties, or other kinds of non-probabilistic knowledge. This in particular implies that resource theories cannot study exactly how lack of knowledge influences the way that agents may exploit resources. The second aspect is that current resource theories cannot model the way that different agents interact and perceive each other's actions. Resource theories are lacking tools to include sev- eral agents within the same framework, who may differ in their knowledge and their language to describe resources, such as a macroscopic observer and a microscopic \demon" in the Maxwell demon thought experiment. The final aspect concerns the understanding of subsystems and the composition of resources, as well as the way that one can quantify resources and the cost of transformations. Current resource theories pursue a bottom-up-approach: individual resources can be composed with the tensor product, and resources are quantified in terms of copies of resources and their asymptotic conversion rates. However, with such a model one cannot capture situations in which there are unknown correlations between subsystems | likewise, resource theories are lacking tools to quantify resources in the single-shot regime. This thesis develops the missing tools to address these issues and formalises subjec- tive knowledge as well as the interaction of different agents in resource theories. vii viii Kurzfassung Ein einfacher und doch vielversprechender Weg, die meisten Probleme der Physik und Informationstheorie anzugehen, ist, zu fragen: \Was ist m¨oglich, und was unm¨oglich, wenn man unter gewissen Restriktionen arbeitet? Und was k¨onnte man erreichen, wenn man noch zus¨atzliche Ressourcen zur Verf¨ugungh¨atte?" Diese Idee liegt den sogenannten Ressourcentheorien zugrunde. Ressourcentheo- rien wurden in der Quanteninformationstheorie als Erstes im Zusammenhang der Verschr¨ankungdurch die LOCC-Theorie betrachtet, welche die ber¨uhmten Ver- schr¨ankungsmasseder `entanglement of formation' und des `distillable entanglement' einf¨uhrte| Ressourcentheorien allgemein werden bis heute in vielen Bereichen mit grossem Erfolg angewandt. Ein Beispiel hierf¨urliefert die Thermodynamik, welche lange schon als eine Theorie verstanden wurde, die uns fundamentale Grenzen setzt, wie etwa das zweite Gesetz der Thermodynamik. Nun konnten Ressourcentheorien f¨urQuanten-Thermodynamik zeigen, dass f¨urmikroskopische Prozesse eine ganze Familie von \zweiten Haupts¨atzen"gelten, die sich mit Hilfe von sogenannten Single- shot Entropien, die wir aus der Informationstheorie kennen, ausdr¨ucken lassen. Obwohl Ressourcentheorien unser Verst¨andnisin vielen Bereichen der Physik und Informationstheorie erweitert haben, gibt es eine Reihe wichtiger Aspekte, die bisher noch nicht explizit von Ressourcentheorien formalisiert wurden oder ¨uberhaupt for- malisierbar waren | diese Aspekte zu beschreiben ist das Hauptziel, und der gr¨osste Beitrag, dieser Dissertation.