Categorical quantum models and logics The work in this thesis has been carried out while the author was employed at the Radboud University Nijmegen, financially supported by the Netherlands Or- ganisation for Scientific Research (NWO) within the Pionier projects “Program security and correctness” during August 2005–August 2007, and “Quantization, noncommutative geometry and symmetry” during August 2007–August 2009. Typeset using LATEX and XY-pic ISBN 978 90 8555 024 2 NUR 910 c C. Heunen / Pallas Publications — Amsterdam University Press, 2009 This work is licensed under a Creative Commons Attribution-No Derivative Works 3.0 Netherlands License. To view a copy of this license, visit: http://creativecommons.org/licenses/by-nd/3.0/nl/. Categorical quantum models and logics een wetenschappelijke proeve op het gebied van de Natuurwetenschappen, Wiskunde en Informatica PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. mr. S.C.J.J. Kortmann, volgens besluit van het College van Decanen in het openbaar te verdedigen op 7 januari 2010 om 13.30 uur precies door Christiaan Johan Marie Heunen geboren op 21 maart 1982 te Nijmegen Promotores: prof. dr. B.P.F. Jacobs prof. dr. N.P.L. Landsman Doctoral thesis committee: prof. dr. S. Abramsky University of Oxford prof. dr. M. Gehrke Radboud University Nijmegen prof. dr. P.T. Johnstone University of Cambridge prof. dr. I. Moerdijk Utrecht University dr. M. Muger¨ Radboud University Nijmegen Preface I cannot allow this thesis to be published without thanking those without whom it could not have been. First and foremost, I am deeply grateful to my promotores. Bart Jacobs, ever cheerful, enthusiastic and ready to explain, taught me the importance of “think- ing with one’s fingers”. From Klaas Landsman I learned the value of “social science”, i.e. how rigorous research can be jump-started by opinions of experts on vague ideas. I am very glad to have had the opportunity to work with such amicable supervisors, and can only hope that their invaluable guidance is re- flected in this thesis. I am also honoured by the effort that the members of the doctoral committee put into reading my work. Especially Peter Johnstone, whose remarks were spot on and reveal a very careful reading, saved me from eternal shame, for which I thank him heartily. In addition to my supervisors, I am indebted to my other co-authors Bas Spit- ters, Ichiro Hasuo, Ana Sokolova and Martijn Caspers for sharing their insight. Especially Bas came up with incomprehensibly many ideas to work out. Fur- thermore, my colleagues in the research community always made me feel very welcome, not only during all those conferences and workshops. In particular, I enjoyed the encouragement, constructive criticisms and advice by John Hard- ing, Isar Stubbe, Jamie Vicary, and Steve Vickers. In the same spirit, I thank Bob Coecke, Marcelo Fiore and Ichiro Hasuo for inviting me on research visits to Oxford, Cambridge and Kyoto, respectively, during which I learned a great deal. Closer to home, I am grateful to all my colleagues at the mathematics depart- ment, who were always ready to answer my probably blatantly obvious ques- tions. The digital security group provided a much appreciated lively atmosphere, despite my research topic not quite fitting in seamlessly. This homely feeling is largely due to Miguel Andres, Łukasz Chmielewski, Flavio Garcia, Ichiro Hasuo, Ron van Kesteren, Gerhard de Koning Gans, Ken Madlener, Peter van Rossum, Ana Sokolova, and Alejandro Tamalet, with all of whom I shared an office over the years. To be fair, it was not only tea and table football: there were certainly also research discussions and reading groups, especially with Ana, Ichiro and v Peter, which I very much appreciated. Lastly, Wojciech Mostowski was always ready to share his intimate knowledge of TEX when mine wasn’t deep enough. The support I enjoyed from outside academia was perhaps just as important, and I would like to thank all my friends for their companionship over the last years, including the (Roman) dinners, movie nights, sailing weekends, snow- boarding trips, and general amusement. Łukasz and Ron in addition accepted the task of being my paranymphs. There are too many more friends to list here, but in particular, the members of karate clubs NSKV Dojo and Shu Ken Ma Shi should be mentioned, as the countless training hours with them provided a lot of fun and relief. Let me conclude with some words of gratitude to my family (in-law) for their care in my development; especially to my brothers, for keeping my feet on the ground, and to my parents, for always being there to help, even with the most practical of things. Finally, and most of all, I thank Lotte, for so much more than can be mentioned here. Utrecht, August 2009 Contents 1 Introduction 1 2 Tensors and biproducts 11 2.1 Examples . 11 2.2 Tensor products and monoids . 14 2.3 Biproducts . 22 2.4 Scalars . 29 2.5 Modules over rigs . 34 2.6 Compact objects . 41 3 Dagger categories 47 3.1 Examples . 47 3.2 Dagger structures . 56 3.3 Quantum key distribution . 66 3.4 Factorisation . 70 3.5 Hilbert modules . 80 3.6 Scalars revisited . 86 3.7 Hilbert categories . 90 4 Dagger kernel logic 99 4.1 Subobjects . 99 4.2 Orthogonality . 106 4.3 Orthomodularity . 111 4.4 Quantifiers . 117 4.5 Booleanness . 127 4.6 Subobject classifiers . 133 5 Bohrification 141 5.1 Locales and toposes . 141 5.2 C*-algebras . 150 vii 5.3 Bohrification . 156 5.4 Projections . 163 5.5 States and observables . 172 Bibliography 181 Index of categories 195 Index of notation 197 Index of subjects 199 Samenvatting 203 viii Chapter 1 Introduction Quantum theory is the best description of nature at very small scales to date. Its principal new features compared to classical physics are superposition of states, noncommutativity of observables, and entanglement. Although strange and coun- terintuitive at first, such features can be exploited once recognised. Entangle- ment, for example, was discovered and regarded as a paradox by Einstein, Podol- sky and Rosen in 1935 [79], but nowadays it is mainly seen as a resource to be used. For example, entanglement enables key distribution protocols, providing each of the participating parties with a string of bits that is guaranteed to be known to them only. Even more so, quantum computers employ entanglement to solve certain problems essentially faster than a classical computer can [173]. To achieve their full potential, such new applications have to come with mathematical proofs. Nobody will use a quantum computer for serious tasks if the programmer cannot vouch for the correctness of the program, and the very attractiveness of quantum key distribution for secret communication lies in the guarantee that there can be no eavesdroppers. Because human intuition is unreliable in the quantum world, a mathematically rigorous way to reason about quantum situations is called for. In other words, we need a logic for quantum physics, and that is what this thesis investigates. Counterintuitive features of quantum physics To illustrate the counterintuitive features of quantum physics, let us first explain the general form of any physical theory. An isolated object is described by the set of states in which it can be, and its empirical properties are modeled by a set of observables, so that a state and an observable can be combined into a real value, modeling the outcome of the act of observation, i.e. measurement. Often, we are not sure about the exact state of an object. Therefore we allow convex combina- 1 Chapter 1. Introduction tions of states. Observation (the pairing of a state with an observable) then only results in a given value with a certain probability. The states about which we have perfect knowledge, i.e. which cannot be written as convex combinations of other states, are called pure states. Finally, there is some way to combine the state spaces of component objects into the state space for a compound object. The above scheme finds a natural home in classical physics, where the pure states of an object form a set X. It might come with a topology or some geo- metric structure, but in principle X is just a set. Observables are functions f : X ! R, such as speed. Observation of an object in a pure state x 2 X results in a sharp observed value f(x). Elementary propositions, like “the ob- ject’s speed is between 10 and 20 m/s”, are dictated to be true in a pure state x, if and only if f(x) 2 (10; 20), i.e. if and only if x 2 f −1(10; 20). The state of a compound object completely determines the states of its component objects: if X and Y are the state spaces of the component objects, then X × Y is the state space of the compound object. The traditional formulation of quantum physics, which is due to John von Neumann [213], also fits the above scheme. The state space of a quantum sys- tem has the structure of a Hilbert space X. That is, X comes with an inner product hx j x0i that signifies the probability amplitude of a transition from state x to state x0. Pure states are unit vectors. Observables are self-adjoint opera- tors f : X ! X. By the spectral theorem, which generalises diagonalisation of matrices, every such operator f corresponds uniquely to a family of (so-called projection) operators e∆ : X ! X for every interval ∆ ⊆ R. In contrast to classi- cal physics, even observation of an object in a pure state only gives probabilistic results.