Automated rewriting for higher categories and applications to quantum theory Krzysztof Bar University College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2016 Acknowledgements First and foremost I would like to express the most immense gratitude to my supervisor Dr Jamie Vicary, whose patience and dedication always went far above and beyond the call of duty. As our weekly meetings often turned into daily discussions, where he taught me about the intricacies of category theory and quantum physics, but most importantly about how to express scientific results in a rigorous, yet accessible fashion. The guidance he provided gave me the sense of security, the importance of which cannot be understated in the life of a research student. Ever since we started working together in the middle of my first year, there has never been a slightest doubt in my mind that our research will lead to a successful completion of my DPhil. At the end of it all, I hope that I kept the promise I gave him during our first meeting, and he saved more time as the result of our collaboration, than he put into teaching me. There are other distinguished academics who I had the fortune of encountering on my path. I want to thank my co-supervisors prof. Bob Coecke and prof. Samson Abramsky, who welcomed me as a member of their research group and initiated the entire research programme of categorical quantum mechanics, which has had such a profound impact on my academic life. My undergraduate college tutor prof. Andrew Ker for giving me a chance to come to Oxford all these years ago to begin the most transformative experience of my entire life. Prof. Artur Ekert, for making it possible for me spend the summer of 2014 at the Centre for Quantum Technologies in Singapore, which was one of the most scientifically prolific periods of my entire DPhil. Prof. John Baez whose guidance and encouragement during the initial stages of my work on higher category theory were invaluable. I would also like to thank Dr Sergusz Trzeciak for his mentorship and advice on career planning. One's academic journey begins much earlier than at the moment they first set a foot in a university lecture theatre, for that reason I want to thank my high school and middle school mathematics teachers Boguslaw Kraszewski, Pawel Walter, Wioletta Kaszuba, for sparking my interest in the subject and teaching me that, in pursuing mathematics, there is no value more important than hard work. I am thankful for the time spent with fellow doctoral students at the Quantum group at the Oxford University Computer Science Department. In particular I want to thank Linde Wester and Dominic Verdon, with whom I shared research interests and ideas on how to survive and thrive in the academic environment. The completion of this thesis and my studies at Oxford in no way would have been possible without the support of a large number of people who I am proud to call my friends. All of them deserve much more than just a short recognition in this thesis, but I hope that this small token of appreciation can begin to repay them for their friendship and the innumerable acts of kindness. Firstly, I want to thank Kuba Redlicki for the eight years spent together in Oxford both on and off the volleyball court and for showing me that both in sports and in life in general `only the next point matters'. Joanna Bagniewska for her friendship and countless hours spent discussing plans on how to improve Poland, the United Kingdom and the world in general. I am not sure whether success here is realistic, but I am certain that we will give it our best shot. And to Robert Pisarczyk for all he has taught me about quantum physics and about science in general, especially during the summer we spent as flatmates in Singapore. I would also like to acknowledge the many friends and collagues at the Oxford University Polish Society and the Federation of Polish Student Societies in the UK. Working and spending time alongside you sometimes felt almost like pursuing a parallel degree in public service. In particular I would like to acknowledge Jan Domanski, Ewelina Gregolinska, Aleksandra Janusz, Maciej Lisik, Mateusz Mazzini, Mikolaj Firlej, Anna Muszkiewicz, Aleksandra Pedraszewska, Marek Przytula, Bartosz Redlicki, Mateusz Ujma, Maria Wilczek and Andrzej Wolniewicz. I thoroughly enjoyed your company during the countless conferences, formal dinners and less formal parties afterwards and I know many more will come in the future. You provided me with welcomed distractions from the sometimes mundane reality of life as a research student and for that I sincerely thank you. Although I am certain that without your (always much appreciated) interference I would have finished this thesis many months ago. Special thanks to Marta Tondera and Anna Bialas, who I had the pleasure of mentoring for a period of time, however in the end I am sure that I have learnt from you at least as much as you did from me. To Magdalena Richter and Tomasz Cebo for the shared interest in popularisation of science and the time we spent together on `Science. Polish Perspectives' building a multi- disciplinary community of researchers. To my friends from the time as an undergraduate at University College, Oxford: Amy Bealey, Michael Captain, Dan Waters. Our reunion meetings in Oxford, London and Poole, as well as the many stimulating conversations always provided me with a much needed respite and allowed me to return to research with a rejuvenated interest. Also to those, especially Joanna Porebska and Przemyslaw Pobrotyn, who proofread the drafts of this thesis and provided many helpful remarks. My years at Oxford would not have been complete without the friends and teammates at the Oxford University Volleyball Club: Michael D'Angeli, Rowan Hamill-McMahon, Alexander Stockenhuber and many others throughout the years. But the most heartfelt thanks go to the 2015/2016 team that won the BUCS Trophy, finished 3rd in the UK in the EVA student Cup and won the Varsity match: Fabio Anza, Andreas Iskra, Gytis Jankevicius, Sanders Lazier, Adam McBride, Christos Mavrokefalos, Stefan Nekovar, David Novotny, Jonas Pollex, Kuba Redlicki, Sven Sabas, Nicolas Stone-Villani and Tilo Zollitsch. The feeling of togetherness and the sense of comradery that we shared throughout the season is something that I will cherish for the rest of my life, especially on the days when I return to the Cape of Good Hope for a Wednesday drink to celebrate yet another victory. I am forever indebted to my friends from home: Jakub Hyla, Michal Kulak, Kamil Morawski, Jan Porebski, Juliusz Sikora, Krzysztof Slupinski, Jan Tarnawski and Milosz Tatarczuch for all the time we spent together and for giving me something to look forward to in the future. But most importantly for, whenever I return to Krakow, making me feel as though I had never left. Finally, I would like to thank my parents Anna and Waldemar, who both sacrificed so much to make it possible for me to pursue the most ambitious goals, I am deeply grateful for their unwavering support and for challenging me to always try to be the best version of myself. Thank you to my brother Piotr for his healthy dose of skepticism and that I can always count on him, also to my grandparents Emilia, Teresa, Bronislaw and Marian for always being there for me. When I first begun my studies of cateory theory, I encountered the following quote by Stefan Banach: A mathematician is a person who can find analogies between theorems, a better mathematician is one who can see analogies between proofs and the best 3 mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies. Myself, I have often struggled to even understand the theorems and never aspired to be anything more than a mathematician. Nonetheless, the experience of being a research student, as deeply humbling as it may have been at times, is one which I will always hold dearest. Thank you, from the bottom of my heart, to all those who made it possible. 4 Abstract This thesis consists of two contributions built on the foundation of higher category theory. The first is a novel framework for rewriting in higher categories. Its theoretical foundation is the theory of quasistrict higher categories and the practical realisation is a proof assistant Globular. Building on this, we propose a new definition of a quasistrict 4-category, and prove a result that in a quasistrict 4-category, an adjunction of 1-morphisms gives rise to a coherent adjunction satisfying the butterfly equations. The second contribution is the application of a higher categorical formalism to quantum theory to show equivalence between mutually unbiased bases and satisfaction of quantum key distribution schemes, and to prove correctness for construction of a particular set of solutions to the Mean King problem. Contents 1 Introduction 1 1.1 Key contributions and thesis outline . .2 1.2 Background on higher categories . .4 1.2.1 Quasistrict n-categories . .5 1.2.2 The need for automation . .6 1.2.3 Periodic table of higher categories . .7 1.3 Proofs in the graphical calculus . .8 1.3.1 Globular . 11 1.4 Signatures and Diagrams . 12 2 Automated rewriting for higher categories 17 2.1 Rewriting systems and presentations of n-categories . 18 2.2 Basic structure . 19 2.2.1 Signatures and diagrams . 20 2.3 Rewriting . 23 2.3.1 Correctness of the rewriting construction . 27 2.4 Composition .