Alternative theories in Quantum Foundations Andr´eO. Ranchin Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics of Imperial College London July 2016 Supervised by Bob Coecke and Terry Rudolph Department of Physics Imperial College London Alternative theories in Quantum Foundations Andr´eO. Ranchin Submitted for the degree of Doctor of Philosophy July 2016 Abstract Abstraction is an important driving force in theoretical physics. New insights often accompany the creation of physical frameworks which are both comprehensive and parsi- monious. In particular, the analysis of alternative sets of theories which exhibit similar structural features as quantum theory has yielded important new results and physical understanding. An important task is to undertake a thorough analysis and classification of quantum-like theories. In this thesis, we take a step in this direction, moving towards a synthetic description of alternative theories in quantum foundations. After a brief philosophical introduction, we give a presentation of the mathematical concepts underpinning the foundations of physics, followed by an introduction to the found- ations of quantum mechanics. The core of the thesis consists of three results chapters based on the articles in the author's publications page. Chapter 4 analyses the logic of stabilizer quantum mechanics and provides a complete set of circuit equations for this sub-theory of quantum mechanics. Chapter 5 describes how quantum-like theories can be classified in a periodic table of theories. A pictorial calculus for alternative physical theories, called the ZX calculus for qudits, is then introduced and used as a tool to depict particular examples of quantum-like theories, including qudit stabilizer quantum mechanics and the Spekkens- Schreiber toy theory. Chapter 6 presents an alternative set of quantum-like theories, called quantum collapse models. A novel quantum collapse model, where the rate of collapse depends on the Quantum Integrated Information of a physical system, is introduced and discussed in some detail. We then conclude with a brief summary of the main results. Acknowledgments Scientific research resembles the activity of an underground explorer who undertakes the arduous task of digging tunnels and constructing elaborate subterranean passages in search of elusive precious minerals. Naturally, this process { which will more often lead to a frustrating conclusion than to the launch of a fruitful enterprise { cannot be undertaken alone. Friends and family provide a pillar of strength which buffers the impact of the inevitable collapse of theoretical caverns. Fortunately, there is only a minute risk of suffocation and it is only in a metaphorical sense that one may end up covered in dirt and trapped in a confined space. Moreover, failure is a far better teacher than success. I have certainly learnt many things in the last few years. It is my pleasure to mention some of the people who have played an essential role throughout my PhD. First of all, I would like to thank my supervisors Bob Coecke and Terry Rudolph for their insightful help and helpful insights, kindly given whenever necessary. Their patience and understanding provided an invaluable catalyst for intellectual development throughout my years of academic study in London and Oxford. I am also much obliged to Sandu Popescu for introducing me to the fascinating world of quantum physics research and encouraging me to pursue further study. I have had the privilege of working with not just one but two exceptional circles of colleagues: the Oxford Quantum group and the Imperial Controlled Quantum Dynamics group. A particular mention should go to Miriam Backens, Raymond Lal, William Zeng, ii ACKNOWLEDGMENTS iii Ross Duncan, Aleks Kissinger, John Selby, Sean Mansfield, Rui Soares Barbosa, Nadish De Silva, Hugo Nava Kopp, Dan Marsden, Jonathan Barrett, Destiny Chen and to the eleven members of the CQD DTC Cohort 3. Mihai-Dorian Vidrighin and Mark Mitchison especially deserve my appreciation for their friendship and for their irreplaceable assistance and advice. A particular word of thanks should go to the directors of the CQD DTC, particularly Danny Segal, whose compassionate words of support provided solace at a most difficult time. I must also mention the staff of the BHOC, without whose effort and care, the present work would not have seen the light of day. Although it goes without saying, I would like to sincerely thank my parents for their constant encouragement as well as for the key role they have played in my academic and personal development. Finally, my deepest gratitude goes to Sylvia, whose thoughtful and unfailing support has provided an invaluable bedrock for any achievement of mine. I acknowledge financial support from the EPSRC. Declaration of Originality: I declare that all the work presented in this thesis is my own or is properly referenced such that the original source is clearly stated. Copyright Declaration: The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. Author's publications 1. A. O. Ranchin, B. Coecke, \Complete set of circuit equations for Stabilizer Quantum Mechanics". Physical Review A, 90, 012109 (2014). 2. A. O. Ranchin \Depicting qudit quantum mechanics and mutually unbiased qudit theories". EPTCS Quantum Physics and Logic 2014, 172 (2014). 3. K. Kremnizer, A. O. Ranchin \Integrated Information-induced quantum col- lapse". Foundations of Physics, 45 (2015). iv Contents Acknowledgments ii Author's publications iv List of Figures viii 1 Introduction 1 2 Background I: Mathematical tools 8 2.1 Set theory . 9 2.1.1 Axiomatic set theory . 9 2.1.2 Relations and functions . 11 2.1.3 Numbers . 12 2.2 Group Theory . 14 2.3 Algebraic structures . 19 2.3.1 Rings, Fields and Galois theory . 19 2.3.2 Linear Algebra and Graph theory . 24 2.4 Topology and Hilbert spaces . 28 2.4.1 Topology . 28 2.4.2 Topological vector spaces . 32 2.5 Category theory . 36 2.5.1 Categories and functors . 36 2.5.2 Limits . 37 2.5.3 Examples of categories . 38 2.5.4 Natural Transformations and adjoints . 39 2.5.5 Categorical quantum mechanics . 41 3 Background II: Quantum theory 47 3.1 Operational theories . 47 3.2 Quantum mechanics introduced . 49 3.2.1 Orthodox postulates . 49 3.2.2 Operational axioms . 51 3.3 Quantum computation . 52 vi 3.3.1 Quantum circuits . 52 3.3.2 Other quantum computation models . 55 3.4 Non locality and Contextuality . 57 3.4.1 Realism and quantum theory . 57 3.4.2 EPR . 57 3.4.3 Bohr response . 58 3.4.4 Hidden variables and Von Neumann's no go theorem . 59 3.4.5 Bell's theorem and the CHSH inequality . 61 3.4.6 Cirelson bound . 63 3.4.7 Popescu Rohrlich boxes . 63 3.4.8 Generalized CHSH inequality . 64 3.4.9 Mermin non-locality . 65 3.4.10 The over-protective seer . 67 3.4.11 Gleason's theorem . 68 3.4.12 Bell corollary of Gleason's theorem . 69 3.4.13 Kochen Specker theorem . 70 3.4.14 Mermin magic square . 71 3.4.15 Leggett-Garg inequality . 72 3.5 Ontological models for quantum mechanics . 73 3.5.1 Examples of ontological models . 74 3.5.2 Spekkens toy theory . 77 3.5.3 Contextuality for ontological models . 79 3.5.4 PBR theorem . 81 3.6 Ontological interpretations of quantum theory . 83 3.6.1 Bohmian mechanics . 83 3.6.2 Many-worlds theory . 84 3.6.3 Collapse models . 86 3.7 Generalized probabilistic theories . 88 3.7.1 Hardy's operational framework . 89 3.7.2 Information theoretic constraints for quantum theory . 91 3.7.3 Information processing in generalized probabilistic theories . 92 4 The logic of Stabilizer Quantum Mechanics 94 4.1 Introduction . 96 4.2 Stabilizer quantum theory . 98 4.3 ZX network . 99 4.4 Completeness of the ZX calculus . 102 4.5 Quantum circuits for the ZX network axioms . 104 4.6 Proof of the Equivalence Lemma . 106 4.7 A complete set of circuit equations for stabilizer quantum mechanics . 116 4.8 Derivation of an equation between stabilizer quantum circuits from the com- plete set . 120 4.9 Reasoning with the ZX network is easier than using the quantum circuit calculus122 4.10 Conclusion . 123 vii 5 A periodic table of quantum-like theories 125 5.1 Introduction . 128 5.2 Explicit models of theories . 130 5.2.1 Qudit stabilizer quantum mechanics . 131 5.2.2 Spekkens toy theory in higher dimensions . 132 5.3 Depicting qudit quantum mechanics and toy models . 135 5.3.1 Derivation of the qudit ZX calculus . 135 5.3.2 The ZX calculus for qudit quantum mechanics: . 144 5.3.3 Mutually unbiased qudit theories . 153 5.3.4 Picturing stabilizer quantum mechanics . 153 5.3.5 Depicting Spekkens-Schreiber toy theory for dits . 159 5.4 A periodic table of quantum-like theories . 166 5.5 Topological Ontological models . 170 5.6 Further work . 174 6 Quantum collapse theories and Quantum Integrated Information 176 6.1 The philosophy of consciousness . 178 6.1.1 History . 178 6.1.2 Philosophical positions . 180 6.1.3 Problems . 181 6.2 Consciousness and Integrated Information . 183 6.3 Calculating the Quantum Integrated Information . 185 6.4 A review of existing quantum collapse theories . 187 6.4.1 Pearle's collapse equation .