Quantum Correlations in Space-Time: Foundations and Applications Tian Zhang Somerville College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Revised Version, 2020 Abstract The absolute/relative debate on the nature of space and time is ongoing for thousands of years. Here we attempt to investigate space and time from the information theoretic point of view to understand spatial and temporal correlations under the relative assumption. Correlations, as a measure of relationship between two quantities, do not distinguish space and time in classical probability theory; quantum correlations in space are well-studied but temporal correlations are not well understood. The thesis investigates quantum correlations in space-time, by treating temporal correlations equally in form as spatial correlations and unifying quantum correlations in space and time. In particular, we follow the pseudo- density matrix formalism in which quantum states in spacetime are properly defined by correlations from measurements. We first review classical correlations, quantum correlations in space and time, to motivate the pseudo-density matrix formalism in finite dimensions. Next we generalise the pseudo-density matrix formulation to continuous variables and general measurements. Specifically, we define Gaussian spacetime states by the first two statistical moments, and for general continuous variables spacetime states are defined via the Wigner function representation. We also define spacetime quantum states in position measurements and weak measurements for general measurement processes. Then we compare the pseudo-density matrix formalism with other spacetime formulations: indefinite causal structures, consistent his- tories, generalised non-local games, out-of-time-order correlation functions, and path integrals. We argue that in non-relativistic quantum mechanics, different arXiv:2101.08693v1 [quant-ph] 21 Jan 2021 spacetime formulations are closely related and almost equivalent via quantum correlations, except path integrals. Finally, we apply the pseudo-density matrix formulation to time crystals. By defining time crystals as long-range order in time, we analyse continuous and discrete time translation symmetry as well as discuss the existence of time crystals from an algebraic point of view. Finally, we summarise our work and provide the outlook for future directions. Quantum Correlations in Space-Time: Foundations and Applications Tian Zhang Somerville College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Revised Version, 2020 “Space and time are the pure forms thereof; sensation the matter.” — Immanuel Kant, Critique of Pure Reason Acknowledgements I find myself very lucky to have the opportunity to study at Oxford and around the world with so many talented mentors and friends, from whom I have learnt so much. I am very grateful for all the help and support they offer and would express my sincere gratitude to all who have made my DPhil days special. First of all, I would like to thank my supervisor, Prof. Vlatko Vedral, for inviting me to his group, for insightful discussions during these years, and for his enthusiasm for physics which always inspires me. I would also like to thank Dr. Tristan Farrow, who is always so kind and thoughtful, like a big elder brother, not only teaches me how to cope with different academic situations, but also cares about my personal development. I am grateful to Prof. Oscar Dahlsten, who has invited me twice to SUSTech in Shenzhen, China, collaborated with me, especially for his patience on teaching me how to write my first paper, as well as continually giving me lots of guidance and feedback. I also thank Dr. Felix Tennie, who has been so patient and helpful to revise my first-year transfer report and all the suggestions on academic writing. I would like to thank Dr. Chiara Marletto, who collaborated with me on my first project on time crystals, for being so helpful and considerate to give me suggestions on research and writing. I would love to thank the rest of the group, including Anupama Unnikrishnan, Nana Liu, Christian Schilling, Davide Girolami, Reevu Maity, Pieter Bogaert, Thomas Elliott, Benjamin Yadin, Aditya Iyer, Jinzhao Sun, Sam Kuypers, David Felce, and Anicet Tibau Vidal. I would also like to thank the quantum group in the department of computer science, especially Prof. Giulio Chiribella and Prof. Jonathan Barrett for insightful discussion, and Prof. Bob Coecke and lots of others for organising interesting talks, conferences and Wolfson foundation discussions, from which I benefited a lot. Then I want to thank Perimeter Institute for Theoretical Physics, in particular for the visiting graduate fellow program. I would love to thank my local host and advisor, Prof. Lucien Hardy, for inviting me to Perimeter, for regular meetings with me to answer my questions and check my progress, for always being so encouraging and telling me to think of big problems. It is a great pleasure to meet Prof. Lee Smolin at Perimeter and hold weekly meetings with him. I learned so many fascinating ideas and deep thoughts from Lee and gradually started to build on my own research taste. Thanks to Prof. Rafael Sorkin as well for being so patient and helpful and explaining to me on quantum measure theory, irreversibility, nonunitary, and lots of sightful conversation. And thanks to Beni Yoshida, for introducing the wonderful world of black hole information paradox to me and collaborating with me on the black hole final state projection proposal. I would also thank Guifre Vidal, for his guidance on organising the session of black hole information paradox in the quantum information workshop in Benasque, Spain. I would also love to thank the whole quantum foundation and quantum information group, Daniel Gottesman, Robert Spekkens, David Schmid, Tobias Fritz, Denis Rosset, Thomas Galley, Nitica Sakharwade, Zi-Wen Liu, Nick Hunter-Jones, and lots of others that I am sorry I cannot list for all, for having lunches and discussions together and always being so friendly and helpful. I had a brilliant time at Perimeter and I am so grateful for everyone there. I thank Prof. Xi Dong, for hosting my visit at Santa Barbara and discussing possible projects on holographic min- and max-entropy. I also want to thank the organisers and participants of Boulder Summer School 2018, for organising such a great quantum information summer school, where I learned so much and broadened my understanding of quantum information science. Thanks to Felix Leditzky, Graeme Smith, Mark M. Wilde for the opportunity to present my work at Rocky Mountain Summit on Quantum Information. Thanks to Hilary Carteret and John Donohue for inviting me to present my work at Institute of Quantum Computing, University of Waterloo. And thanks to Prof. Otfried Guehne for inviting me to visit and work with his group at University of Siegen. I am very grateful to Prof. Renato Renner at ETH Zurich and Prof. Simon Benjamin at Oxford Materials for being my examiners and all the discussion and suggestions for my thesis. I would thank my valuable friends at Oxford, at Perimeter Institute, at Boulder Summer School, at different conferences, from my undergraduate, Peking University and even long time before. I wish I could list all their names here but I am afraid there are too many of them and I am more afraid I may miss some of their names. Thank you so much, my dearest mum and dad. Thanks to all my family members for their love and understanding. I have had very difficult time during my DPhil. I am so grateful that I have received so much love and support to get through all the hard time. I would express my gratitude, again to so many people along the way for their companion. And thank you, my reader, for taking your time to have a look at this thesis. Abstract The absolute/relative debate on the nature of space and time is ongoing for thousands of years. Here we attempt to investigate space and time from the information theoretic point of view to understand spatial and temporal correlations under the relative assumption. Correlations, as a measure of relationship between two quantities, do not distinguish space and time in classical probability theory; quantum correlations in space are well-studied but temporal correlations are not well understood. The thesis investigates quantum correlations in space-time, by treating temporal correlations equally in form as spatial correlations and unifying quantum correlations in space and time. In particular, we follow the pseudo-density matrix formalism in which quantum states in spacetime are properly defined by correlations from measurements. We first review classical correlations, quantum correlations in space and time, to motivate the pseudo-density matrix formalism in finite dimensions. Next we generalise the pseudo-density matrix formulation to continuous variables and general measurements. Specifically, we define Gaussian spacetime states by the first two statistical moments, and for general continuous variables spacetime states are defined via the Wigner function representation. We also define spacetime quantum states in position measurements and weak measurements for general measurement processes. Then we compare the pseudo-density matrix formalism with other spacetime formulations: indefinite causal structures, consistent histories, generalised non-local games, out-of-time-order correlation functions, and path integrals. We argue that in non-relativistic quantum mechanics, different spacetime formulations are closely related and almost equivalent via quantum correlations, except path integrals. Finally, we apply the pseudo-density matrix formulation to time crystals. By defining time crystals as long-range order in time, we analyse continuous and discrete time translation symmetry as well as discuss the existence of time crystals from an algebraic point of view. Finally, we summarise our work and provide the outlook for future directions. Contents 1 Introduction 1 2 Quantum correlations in space-time 6 2.1 Classical correlations . .8 2.1.1 Correlations in probability theory .