Non-classicality as a computational resource Lorenzo Catani A thesis submitted to University College London for the degree of Doctor of Philosophy Department of Physics and Astronomy University College London October 19, 2018 I, Lorenzo Catani 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. 1 Non-classicality as a computational resource Lorenzo Catani Doctor of Philosophy of Physics University College London Prof. Dan Browne, Supervisor Abstract One of the main questions in the field of quantum computation is where the quantum computational speed-up comes from. Recent studies in the field of quantum foundations have suggested which are the features to be considered as inherently non-classical. One of the major contributions in this direction comes from a result known as Spekkens' toy theory, which is a model built to reproduce quantum theory as a classical phase-space-inspired theory with restric- tions on what an observer can know about reality. The model reproduces many of the features of quantum mechanics, but it does not reproduce non-locality and contextuality. In this thesis we first complete Spekkens' toy theory with measurement update rules and a mathematical frame- work that generalises it to systems of any finite dimensions (prime and non-prime). We also extend the operational equivalence between the toy theory and stabilizer quantum mechanics to all odd dimensions via Gross' Wigner functions. We then use the toy theory to represent the non-contextual and classically simulatable part of the computation in state-injection schemes of quantum computation where contextuality is a resource. In the case of qubits, we show that the subtheories of quantum mechanics represented in the toy model can achieve the full stabilizer theory via state-injection and we associate different proofs of contextuality to different injection processes. Stepping back from Spekkens' toy theory, we conclude by focusing on single system protocols that compute non-linear functions (similarly to the popular CHSH game) which show quantum advantages even in absence of non-locality and contextuality (in its standard notions). We analyse their performances (formalised in Bell's and Tsirelson's bounds) in relation to Lan- dauer's principle, which associates entropic costs to irreversible computations, and to a new notion of contextuality for sequences of transformations. 2 Impact Statement We are probably living in the golden age of the foundations of quantum mechanics. Many re- search groups originated all over the world, conferences are organised every year and researchers do not have to hide with embarrassment while studying what is the nature of quantum reality. This is due, in addition to the universally fascinating questions charaterising the field, to the concrete applications that have resulted from these studies. The main example is Bell's theo- rem, stating that quantum mechanics is incompatible with the intuitive local realism underlying classical physics. It can be stated as the violation of a simple inequality, that has been proven to guarantee stronger cryptographic security and to certify whether numbers are intrinsically random (crucial matter, for example, for gambling companies). Much has still to be discovered and other applications, in particular in the directions of developing more powerful computers, are around the corner. The results of this thesis, mainly concerning contextuality { a notion originating from a similar result to Bell's theorem { contribute to the understanding of what are the resources that justify why quantum computers are more powerful than classical ones. As such, they are beneficial towards the big goal of developing a quantum computer. The impact of this discovery would be huge as, for example, universal quantum computers would be able to break current cryptographic techniques that guarantee the security of our financial transactions. We also deal with examples of quantum computational advantages that involve single systems only, thus extending the area of application and impact of our results. Moreover, some of our mathematical techniques, in particular when dealing with systems of non-prime dimensions, will hopefully provide a new methodology to treat models of computation based on more complex systems than the usual two-level systems (bits/qubits). We believe that extending the knowledge on 3 the resources for quantum technologies would be a benefit in the academical environment, as it would trigger new research projects, open the possibility of new faculty courses and be one of the leading topics in top-level scientific journals. Lastly, understanding how the world works in light of the mysterious and bizarre phenomena described by quantum mechanics is already an extremely frequent theme in popular-science magazines, and our results would fit well in that context too. 4 List of Publications and Preprints The work presented in this thesis contains material from the following publications: 1. L. Catani and D. E. Browne, Spekkens' toy model and its relationship with stabiliser quantum mechanics, New journal of physics, vol. 96, no. 5, p. 052112, 2017. 2. L. Catani, N. D. Silva, and D. E. Browne, Spekkens' toy model in quantum computation and contextuality as a resource, IOS Press Amsterdam, SIF Bologna: accepted for pub- lication in Proceedings of the International School of Physics Enrico Fermi, course 197, foundations of quantum theory, edited by E. M. Rasel, W. P. Schleich and S. Woelk ed., 2017. Other preprints by the author are: 1. L. Catani and D. E. Browne, State-injection schemes of quantum computation in Spekkens' toy theory, arXiv:1711.08676 [quant-ph], 2017. Accepted for publication in Physical Re- view A. 2. L. Henaut, L. Catani, D. E. Browne, S. Mansfield, and A. Pappa, Tsirelson's bound and Landauer's principle in a single-system game, arXiv:1806.05624 [quant-ph], 2018. 5 Acknowledgments Many people deserve to be thanked for allowing me to achieve this important goal. First of all, I am indebted to my supervisor, Dan Browne, for his expertise, patience and enthusiasm. For motivating me with his illuminating ideas and subtle comments. For knowing when it was time to leave me the independence, both to fail and to create, and for encouraging me when I was stuck. I have always had the constant feeling that I could rely on him and I could have not asked for a better guide. My examiners, Jonathan Barrett and Lluis Masanes, need to be thanked for their careful reading of this thesis. Their insightful comments significantly improved this document. Any remaining inaccuracies result from me. Besides my supervisor and examiners, I would like to thank the inspiring figures that allowed me to undertake my placement in Canada last year and start stimulating research projects with them: Robert Raussendorf for his knowledge and always inviting me to be part of his research group and Robert Spekkens for feeding my passion for quantum foundations with extremely motivating and deep conversations. I am also sincerely grateful to their research groups and close collaborators for widening my research from various perspectives, in particular Juan Bermejo- Vega, Piers Lillystone, Hammam Qassim and David Schmid. I also thank Matt Leifer for giving me the possibility to join his group in California, allowing me to follow my passion for foundational questions and work on ideas that I find inspiring and promising. I want to express my sincere gratitude to the CDT in its entirety, starting from cohort one { Andrew, Cameron, Carlo, Claudia, Johnnie, Josh, Sherif, Tim and Tom { a group of friends I shared my academic pains and joys with over the last four years. I cannot forget Lopa for always being there and ready to cheer us up, whatever the issue. I also thank my second 6 supervisor Simone Severini for the never banal conversations and his humanity. I sincerely thank the lads in office C25. Despite sharing a cage with no natural lights and fresh air, I have to admit that it has been the best place to spend my PhD years. Every joke, lunch and secret Santa made my days sweeter than expected. Carlo, both Toms, Alex and Mike, you will be missed. The thanks has to be extended to the whole circle of friends in the physics department here at UCL and my research group for being part of a delightful environment in which to do research. Lastly, I owe a debt of gratitude to my friends from home, here in London, the ones back in Italy, and, more importantly, my family for always supporting me. I feel privileged to have you at my side. You have always been present in the bad moments and ready to celebrate my achievements that would not have been possible otherwise. 7 Contents 1 Introduction 13 1.1 Overview........................................ 13 1.2 Structure and outline of the thesis.......................... 20 2 Spekkens' toy theory in all dimensions and its relationship with stabilizer quantum mechanics 27 2.1 Background....................................... 31 2.1.1 Spekkens' theory................................ 31 2.1.2 Stabilizer quantum mechanics......................... 38 2.1.3 Wigner functions................................ 41 2.2 Update rules - prime dimensional case........................ 47 2.2.1 Adding and removing generators to/from V................. 47 2.2.2 Measurement update rules........................... 49 2.3 Update rules - non prime dimensional case..................... 54 2.3.1 Coarse-graining and fine-graining observables................ 55 2.3.2 Measurement update rules........................... 59 2.4 Equivalence of Spekkens' theory and stabilizer quantum mechanics in all odd dimensions....................................... 62 2.4.1 Stabilizer quantum mechanics - update rules................ 63 2.4.2 Gross' Wigner functions - update rules.................... 65 2.5 Discussion........................................ 70 2.6 Conclusion....................................... 73 8 3 State-injection schemes of quantum computation in Spekkens' toy theory 75 3.1 Background....................................... 78 3.1.1 State-injection schemes of quantum computation.............