Causal Loops Logically Consistent Correlations, Time Travel, and Computation Doctoral Dissertation submitted to the Faculty of Informatics of the Università della Svizzera italiana in partial fulfillment of the requirements for the degree of Doctor of Philosophy presented by Ämin Baumeler under the supervision of Prof. Stefan Wolf March 2017 Dissertation Committee Prof. Antonio Carzaniga Università della Svizzera italiana, Switzerland Prof. Robert Soulé Università della Svizzera italiana, Switzerland Prof. Caslavˇ Brukner Universität Wien, Austria Prof. William K. Wootters Williams College, USA Dissertation accepted on 30 March 2017 Prof. Stefan Wolf Research Advisor Università della Svizzera italiana, Switzerland Prof. Walter Binder and Prof. Michael Bronstein PhD Program Director i I certify that except where due acknowledgement has been given, the work pre- sented in this thesis is that of the author alone; the work has not been submitted previ- ously, in whole or in part, to qualify for any other academic award; and the content of the thesis is the result of work which has been carried out since the official commence- ment date of the approved research program. Ämin Baumeler Lugano, 30 March 2017 ii To this dedication iii iv To my parents, Hanspeter and Zohra v In some remote corner of the sprawling universe, twinkling among the countless solar systems, there was once a star on which some clever animals invented knowledge. It was the most arrogant, most mendacious minute in world history, but it was only a minute. After nature caught its breath a little, the star froze, and the clever animals had to die. And it was time, too: for although they boasted of how much they had come to know, in the end they realized they had gotten it all wrong. They died and in dying cursed truth. Such was the species of doubting animal that had invented knowledge. Peacerich Nothing vi In irgendeinem abgelegenen Winkel des in zahllosen Sonnensystemen flimmernd ausgegossenen Weltalls gab es einmal ein Gestirn, auf dem kluge Tiere das Erkennen erfanden. Es war die hochmütigste und verlogenste Minute der ‘Weltgeschichte’; aber doch nur eine Minute. Nach wenigen Atemzügen der Natur erstarrte das Gestirn, und die klugen Tiere mußten sterben. Es war auch an der Zeit: denn ob sie schon viel erkannt zu haben sich brüsteten, waren sie doch zu letzt, zu großer Verdrossenheit, dahinter gekommen, daß sie alles falsch erkannt hatten. Sie starben und fluchten im Sterben der Wahrheit. Das war die Art dieser verzweifelten Tiere, die das Erkennen erfunden hatten. Friedrich Nietzsche vii viii Abstract Causal loops are loops in cause-effect chains: An effect can be the cause of that ef- fect’s cause. We show that causal loops can be unproblematic, and explore them from different points of view. This thesis is motivated by quantum theory, general relativity, and quantum grav- ity. By accepting all of quantum theory one can ask whether the possibility to take superpositions extends to causal structures. Then again, quantum theory comes with conceptual problems: Can we overcome these problems by dropping causality? Gen- eral relativity is consistent with space-time geometries that allow for time-travel: What happens to systems traveling along closed time-like curves, are there reasons to rule out the existence of closed time-like curves in nature? Finally, a candidate for a theory of quantum gravity is quantum theory with a different, relaxed space-time geometry. Motivated by these questions, we explore the classical world of the non-causal. This world is non-empty; and what can happen in such a world is sometimes weird, but not too crazy. What is weird is that in these worlds, a party (or event) can be in the future and in the past of some other party (time travel). What is not too crazy is that this theoretical possibility does not lead to any contradiction. Moreover, one can identify logical consistency with the existence of a unique fixed point in a cause-effect chain. This can be understood as follows: No fixed point is the same as having a contradiction (too stiff), multiple fixed points, then again, is the same as having an unspecified system (too loose). This leads to a series of results in that field: Characterization of classical non-causal correlations, closed time-like curves that do not restrict the actions of experimenters, and a self-referential model of computation. We study the computational power of this model and use it to upper bound the computational power of closed time-like curves. Time travel has ever since been term weird, what we show here, however, is that time travel is not too crazy: It is not possible to solve hard problems by traveling through time. Finally, we apply our results on causal loops to other fields: an analysis with Kol- mogorov complexity, local and classical simulation of PR-box correlations with closed time-like curves, and a short note on self-referentiality in language. ix x Acknowledgements Danke an alle! A giant “THANK YOU” goes to Stefan; thank you for the support in so many aspects, from the most professional to the most personal. This time span of four and a half years I spent with you in Lugano and Gandria highly influenced me in a great way. Thank you for all the long discussions at the university, at home, in the car, in the train, in the lake, and in bars. Thank you for all these opportunities to discuss our work with other researchers, to write articles, etc. Thank you for your generosity. I also thank all members of our research group: Alberto, Arne, Julien, Pauli, Vroni — what a time!, and Caslavˇ Brukner who bared our visits in Vienna an uncountable number of times, and for all help and discussions. I thank Adarsh “Adu” Amirtham, Mateus Araújo, Christian “Badi” Badertscher, Kfir Barhum, Tomer Barnea, Charles Bédard, Claus Beisbart, Saif Ben Bader, Boyan Beronov, Matteo Biondi, Jo Bowles, Cyril Branciard, Gilles Brassard, Anne Broadbent, Harry Buhrmann, Antonio Carzaniga, Giulio Chiribella, Fabio Costa, Claude Crépeau for giv- ing us the opportunity to talk in Barbados, Borivoje Dakic, Raffaele De Vecchi, Leonardo Disilvestro, Elisabeth Dürr for enduring this process and the endless discussions, her family, Helen Ebbe, Aryan Eftekhari, Adrien Feix, Jürg Fröhlich for his curiosity and dis- cussions, and Eva, Christina Giarmatzi, Manuel Gil, Alexei Grinbaum, Philippe Guérin, my father “Paa” Hanspeter, Marcus Huber, Robert Jonsson, Yeong-Cherng Liang, Damian Markham, Roger Müller, Ognyan Oreshkov, Kiryl Pakrouski, Dimosthenis Pasadakis, Paolo Perinotti, Marcel Pfaffhauser, Thierry “miboï” Pirrolet, Tim Ralph, Sandra Rankovic, Jibran Rashid, Renato Renner, my sister Rim and her son Gabriel, Benno Salway, Ran- dolf Schärfig, Martin Schüle, Sacha Schwarz, Robert Soulé, André Stefanov, Thomas Strahm, Naïri Usher, William “Bill” Wootters, my mother “Yaa” Zohra, Magdalena Zych, everyone from the decanato in Lugano, and everyone else that supported me. Finally, I thank the Reitschule Bern, Rössli Bar, Kreissaal Bar, Drei Eidgenossen, Café Bar Rosenkranz, Café LaDiva, Bar Oops, Bar Tra, La Folie en Tête, Café Lassa, and everyone of the Facoltà Indipendente di Gandria. xi xii Contents Contents xiii 1 Introduction 1 1.1 Motivation and consequences . .2 1.2 Antinomies . .4 1.3 Main results and outline . .5 2 Motivations 7 2.1 Quantum theory: EPR correlations . .8 2.1.1 The Einstein-Podolsky-Rosen argument . .8 2.1.2 Local realism and Reichenbach’s principle . .9 2.1.3 No common cause . 11 2.1.4 No direct causation . 14 2.1.5 Saving local realism: Relative, retro, and emergent causality . 17 2.2 Relativity theory: Closed time-like curves . 21 2.2.1 History . 22 2.2.2 Gödel on closed time-like curves . 25 2.3 Quantum gravity: Hardy’s approach . 26 3 On causality 29 3.1 Definition of causal relations . 30 3.1.1 Freeness from space-time relations and correlations . 31 3.1.2 Causal relations from freeness and correlations . 31 3.2 Causal correlations and causal inequalities . 33 3.3 Causal loops . 35 3.3.1 Antinomies . 36 4 Correlations without causal order 39 4.1 Assumptions . 40 4.2 Mathematical framework . 41 4.2.1 Examples of classical processes . 43 4.2.2 Logical consistency . 45 xiii xiv Contents 4.3 Interpretation of the environment . 46 4.4 Characterization with polytopes . 47 4.4.1 Polytope of logically consistent environments . 49 4.4.2 Deterministic-extrema polytope . 50 4.4.3 Polytopes with binary systems for one to three parties . 50 4.5 Characterization with fixed points . 56 4.5.1 Illustrations of the fixed-point theorems . 58 4.6 Non-causal correlations . 60 4.6.1 Probabilistic non-causal correlations . 60 4.6.2 Deterministic non-causal correlations . 62 4.7 Reversible environments . 63 4.7.1 Reversible environments from the deterministic-extrema polytope 65 4.7.2 Environments from outside of the deterministic-extrema poly- tope cannot be made reversible . 67 4.7.3 Necessity of some source and some sink . 67 4.7.4 Example of a reversible non-causal environment . 69 4.8 Quantum correlations without causal order . 69 4.8.1 Framework . 70 4.8.2 Reversible non-causal process matrix . 72 4.8.3 The classical framework arises as limit of the quantum framework 74 4.9 Discussion . 75 5 Closed time-like curves 77 5.1 Closed time-like curves in general relativity . 78 5.2 Logical and physical principles . 80 5.3 Logically consistent closed time-like curves . 82 5.3.1 The model . 83 5.3.2 Reversibility . 85 5.3.3 Example of a logically consistent closed time-like curve . 87 5.4 Other models of closed time-like curves . 89 5.4.1 Deutschian closed time-like curves .