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Categorical Duality in Probability and Quantum Foundations

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Categorical Duality in Probability and Quantum Foundations Categorical Duality in Probability and Quantum Foundations Proefschrift ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. dr. J.H.J.M. van Krieken, volgens besluit van het college van decanen in het openbaar te verdedigen op dinsdag 3 oktober 2017 om 10.30 uur precies door Robert William Johnston Furber geboren op 10 augustus 1986 te Londen, Verenigd Koninkrijk. Promotor: Prof. dr. B.P.F. Jacobs Manuscriptcommissie: Prof. dr. P. Busch (Universiteit van York, Verenigd Koninkrijk) Prof. dr. N.P. Landsman Prof. dr. M. Ozawa (Nagoya Universiteit, Japan) Prof. dr. P. Panangaden (McGill Universiteit, Montreal, Canada) Prof. dr. A. Simpson (Universiteit van Ljubljana, Sloveni¨e) Categorical Duality in Probability and Quantum Foundations Doctoral thesis to obtain the degree of doctor from Radboud University Nijmegen on the authority of the Rector Magnificus prof. dr. J.H.J.M. van Krieken, according to the decision of the Council of Deans, to be defended in public on Tuesday October 3, 2017 at 10.30 hours by Robert William Johnston Furber born August 10, 1986 in London, United Kingdom. Supervisor: Prof. B.P.F. Jacobs Manuscript committee: Prof. P. Busch (University of York, UK) Prof. N.P. Landsman Prof. M. Ozawa (Nagoya University, Japan) Prof. P. Panangaden (McGill University, Montreal, Canada) Prof. A. Simpson (University of Ljubljana, Slovenia) Dedicated to my parents, Rosemary Johnston and James Furber. Contents Contents i Introduction 3 0 Preliminaries 7 0.1 Convexity in Vector Spaces . 7 0.2 Ordered Vector Spaces . 20 0.3 Dualities, Polars and Bipolars . 23 0.3.1 Polars . 24 0.4 Category Theory . 29 0.4.1 Monads . 31 1 C∗-algebras, Probability and Monads 43 1.1 Introduction . 43 1.2 Preliminaries on C∗-algebras . 46 1.2.1 Effect modules . 52 1.3 Set-theoretic Computations in C∗-algebras . 58 1.4 Discrete Probabilistic Computations . 61 1.5 Continuous Probabilistic Computations . 64 1.6 The Category of Markov Kernels . 70 2 Base-Norm Spaces 87 2.1 Introduction . 87 2.2 Definitions . 88 2.2.1 Base-Norm Spaces . 88 2.2.2 Bounded Convex Sets . 96 2.2.3 Comparison of Definitions . 101 2.3 Relationship to C∗ and W∗-algebras . 104 i CONTENTS 1 2.4 Relationship to Monads . 106 2.4.1 The Base and Subbase Functors and Their Left Adjoints 109 2.4.2 Bounded Affine Functions and the Dual Space . 129 2.5 Dualities with Order-Unit Spaces . 137 2.5.1 The Dual Adjunction . 137 3 Smith Spaces 145 3.1 Introduction . 145 3.2 Smith Spaces . 147 3.2.1 β and σ as functors . 166 3.3 Compact Convex Sets and Smith Base-Norm Spaces . 172 3.3.1 Continuous Affine Functions and the Strong Dual . 175 3.4 Compact Effect Modules and Smith OUSes . 182 3.4.1 Relationship to Convex Sets . 192 3.5 Universal Enveloping Objects . 200 3.6 Relationship to C∗ and W∗-algebras . 203 4 Compact Convex Sets, R and E 209 4.1 Introduction . 209 4.2 Swirszcz's´ Theorem for R . 210 4.3 Swirszcz's´ Theorem for E . 218 4.4 Compact Effect Modules . 224 4.5 Closing Remarks . 225 A Miscellanea 227 A.1 Elementary Real Analysis . 227 A.2 General Topology . 228 A.3 Absolutely Convex Sets . 230 A.4 Effect Modules . 231 A.5 Order-Unit Spaces . 232 A.6 Asimow's Example . 235 B Summary 245 C Samenvatting 247 D Index of Categories 249 Index 254 Bibliography 257 2 CONTENTS Introduction When we consider the semantics of programs, we can consider state transform- ers and predicate transformers.A state transformer describes the action of the program, taking its initial state to its final state. Predicate transform- ers go in the opposite direction and describe how to take a predicate to its weakest precondition. In the classical case of nondeterminism [96, 114] an iso- morphism between state transformers and predicate transformers is obtained by restricting predicate transformers to those that are healthy [114]. This is the reason for our focus on categorical dualities, as the relationship between state and predicate transformers is best expressed as a contravariant equivalence of categories. In general, as well as a category of predicates and a category of states, there is also a category of computations. Together, these form a state-and- effect triangle [56]. That is to say, we have functors between these categories as follows: Predicatesop > - States i n 7 Computations such that the diagram commutes up to isomorphism. In general, the definition of a state-and-effect triangle requires only that the two functors at the top form an adjunction [56], but we aim to get an equivalence. State-and-effect triangles appear in [52], and are considered in detail in [57]. In the first chapter, we concern ourselves with probabilistic computation. ∗ op We prove that K`(R) ' CC AlgPU, which can be considered to be a prob- abilistic generalization of Gelfand duality. We also give a brief introduction to effect modules, their relationship to order-unit spaces, and the expectation monad. 3 4 INTRODUCTION Chapters two to four can be considered to form, taken together, a non- commutative generalization of chapter one. In fact, we move to a setting that not only includes non-commutative C∗-algebras, but also situations in which a multiplication cannot even be defined. The convex set D(X) of all finitely supported probability distributions can be considered to be the \state space" of some system, with elements of X, or their corresponding Dirac distributions, being the \pure states", and D(X) being obtained by forming \mixed states", where convex combinations correspond to randomized mixtures. In quantum mechanics, one considers the convex set of density matrices DM(H), on a Hilbert space H, as a state space. Again, the way of interpreting randomized probabilistic mixtures of states is by forming convex combinations. However, this time there is a difference { in D(X) each mixed state is expressible uniquely as a convex combination of pure states, but this does not hold for DM(H) when dim H ≥ 2. The convex structure of DM(H) can be considered to be the \probabilistic part" of quantum mechanics. One can then extend this idea to interpret convex sets, of various kinds, as state spaces in some generalized theory of probability. This is the viewpoint of generalized probabilistic theories [23, 31, 47, 91, 123, 9, 12, 11]. The reason we try to find a state-and-effect triangle for the category of non- commutative C∗-algebras is that quantum programs can always be represented using morphisms between non-commutative C∗-algebras. We might only use, for example, finite-dimensional C∗-algebras, or only those of the form B(H), n or only B(C2 ), but as long as we can make a state-and-effect triangle for C∗-algebras, we can restrict it to obtain one for these cases. In the second chapter, we give an introduction to base-norm spaces and their relationship to convex sets. This is mainly for application in the next chapter. Base-norm spaces are intended as a way of \freely" producing an ordered vector space for a convex set to live inside. We have tried to clear up any confusion regarding inequivalent definitions of base-norm space that are in use by various authors, comparing several of these definitions to each other with explicit examples, and introducing the term pre-base-norm space for the weakest definition in common use. We give a proof of the known fact that every bounded convex set can be embedded as the base of a pre- base-norm space (making an equivalence PreBNS ' BConv), and use this to slightly generalize a result from the literature to show that sequentially complete bounded convex sets are exactly those convex sets embeddable as bases of Banach base-norm spaces (so BBNS ' CBConv). We then give an adjunction between base-norm and order-unit spaces, based on the duality between states and effects, and restrict this to an equivalence in the standard way. Then we briefly discuss how this equivalence is not quite adequate as a 5 generalization of the commutative case in chapter one. In the third chapter, we first introduce a new characterization of Akbarov's Smith spaces and their duality with Banach spaces. From this we can produce two dualities involving base-norm and order-unit spaces, depending on whether one takes the base-norm spaces or the order-unit spaces to be Smith spaces. If we take the base-norm spaces to be Smith, we get a state-and-effect triangle where the category of computations is C∗-algebras, which gives us one possible state-transformer and predicate-transformer pair of semantics for quantum programs. In fact, the state-transformer semantics corresponds to what is called the Schr¨odingerpicture, and the predicate transformer semantics to the Heisenberg picture. We also produce another state-and-effect triangle, having the category of W∗-algebras and normal maps as computations, where the order-unit spaces are Smith. We show in each case (whether it is C∗ or W∗-algebras) how to turn the base-norm and order-unit space equivalences into equivalences between a category of convex sets (CBConv or CCL) and a category of effect modules (CEMod and BEMod, respectively). We would prefer, in fact, to prove the dualities directly in this setting, but we could only do so using the extra facilities available in the vector space setting (linear independence, locally convex topologies, and the Hahn-Banach and bipolar theorems). These dualities give two generalizations of the duality between states and effects in [24, §3.4] to the infinite-dimensional case, although with the drawback that we consider only positive maps and not completely positive ones.
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