# Samson Abramsky, University of Oxford

**Categorical quantum mechanics: The "monoidal" approach**

January 2008, *Categories, Logic and the Foundations of Physics I*

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**Information flow in physics, geometry, logic and computation V: A tale of dependence and separation**

March 2008, *Clifford Lectures 2008*

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# Howard Barnum, Los Alamos National Laboratory

**(No title)**

March 2008, *Logic, Physics and Quantum Information Theory*

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# John Barrett, University of Nottingham

**Knots and links in braided quantum field theory**

August 2008, *Categories, Logic and the Foundations of Physics III*

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I will explain some aspects of braided quantum field theory: how it is a generating function of knots and links, and also a little about the relation to 3d quantum gravity.

# Bruce Bartlett, University of Sheffield

**Aspects of duality in 2-categories**

May 2008, *Categories, Logic and the Foundations of Physics II*

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The notion of ‘duality’ plays an important role in quantum algebra and topological quantum field theory, as has been particularly emphasized by Baez and Dolan. One aspect of this is the idea of duals for morphisms in a 2-category, which is a generalization of the idea of rigidity in monoidal categories. I will introduce the notion of an ‘even-handed structure’ on a 2-category as a coherent means of turning right adjoints into left adjoints, and explain how this works in various examples such as fusion categories, braided monoidal categories, 2-Hilbert spaces and derived categories having a ‘trivial Serre functor’.

# Luca Bernardinello, University of Milan-Biocca

**On orthomodular posets generated by transition systems**

July 2008, *Quantum Physics and Logic 2008*

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# Paolo Bertozzini, Thammasat University

**Categories of spectral geometries**

May 2008, *Categories, Logic and the Foundations of Physics II*

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In A. Connes' non-commutative geometry, "spaces" are described "dually" as spectral triples. We provide an overview of some of the notions that we deem necessary for the development of a categorical framework in the context of spectral geometry, namely: (a) several notions of morphism of spectral geometries, (b) a spectral theory for commutative full C*-categories, (c) a tentative definition of strict-n-C*-categories, (d) spectral geometries over C*-categories. If time will allow, we will speculate on possible applications to foundational issues in quantum physics: categorical covariance, spectral quantum space-time and modular quantum gravity.

# Edward Blakey, University of Oxford

**Computational complexity in non-Turing models of computation: the what, the why and the how**

July 2008, *Quantum Physics and Logic 2008*

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# Dan Browne, University College London

**Measurement-based quantum computation**

March 2008, *Logic, Physics and Quantum Information Theory*

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# Bob Coecke, University of Oxford

**A survey of categorical quantum mechanics**

March 2008, *Logic, Physics and Quantum Information Theory*

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**Tutorial: the quantum formalism**

August 2008, *Categories, Logic and the Foundations of Physics III*

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**Classical versus quantum — in pictures**

March 2008, *Clifford Lectures 2008*

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# Louis Crane, Kansas State University

**Relational topology and quantum gravity**

January 2008, *Categories, Logic and the Foundations of Physics I*

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We explore arguments for replacing the absolute point set by a sheaf over the site of observation as a foundation for quantum gravity. Time permitting, we consider apparent geometry as a formulation for relational geometry.

**Model categories in quantum gravity**

August 2008, *Categories, Logic and the Foundations of Physics III*

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We examine a plausible physical hypothesis which would allow us to model quantum regions as differential graded Hopf algebras or as differential graded categories.

# Yannick Delbecque, McGill University

**Quantum higher types**

July 2008, *Quantum Physics and Logic 2008*

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# Ellie D'Hondt, Free University of Brussels

**Classical knowledge for quantum security**

July 2008, *Quantum Physics and Logic 2008*

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# Andreas Doering, Imperial College London

**Tutorial: conceptual issues in quantum theory**

August 2008, *Categories, Logic and the Foundations of Physics III*

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# Fay Dowker, Imperial College London

**Dynamical logic**

May 2008, *Categories, Logic and the Foundations of Physics II*

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Despite the high regard in which physicists hold General Relativity, the spacetime nature of reality has not yet fully been taken to heart in addressing the question of the interpretation of quantum mechanics. Partial progress was made by Dirac and Feynman by casting the dynamical content of quantum theory in terms of a Sum Over (spacetime) Histories (SOH). Recently it has been suggested that this SOH is part of an interpretive framework in which the rules of inference that are used to reason about physical reality are themselves subject to dynamical law. Just as General Relativity showed that geometry is not fixed and absolute, so Quantum Mechanics may be telling us that “logic” is not a fixed background but part of physics.

# Ross Duncan, University of Oxford

**Classical structures, MUBs, and pretty pictures**

January 2008, *Categories, Logic and the Foundations of Physics I*

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**The logic of complementary observables**

March 2008, *Clifford Lectures 2008*

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**Interacting quantum observables**

July 2008, *ICALP 2008*

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# Bill Edwards, University of Oxford

**Toy quantum categories**

July 2008, *Quantum Physics and Logic 2008*

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# Chris Fewster, University of York

**The locally covariant approach to quantum field theory in curved spacetimes**

August 2008, *Categories, Logic and the Foundations of Physics III*

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A long-established guiding principle for quantum field theory in curved spacetimes is that it should be formulated in a local and geometrically covariant way. Recently, Brunetti, Fredenhagen and Verch have given an elegant formulation of this principle in categorical terms: a theory should be understood as a functor from a category of spacetimes to a category of star-algebras. This conceptual clarification is at the heart of a number of important results for QFT in general spacetimes, such as a proof of the spin-statistics theorem and the perturbative construction of interacting QFT. This talk reviews the locally covariant approach and some of its applications, and also describes joint work with Verch that refines and sharpens the framework.

# Jonathan Grattage, University of Grenoble

**QML in 15 minutes**

July 2008, *Quantum Physics and Logic 2008*

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# Keith Hannabuss, University of Oxford

**Categories and non-associative C*-algebras in quantum field theory**

August 2008, *Categories, Logic and the Foundations of Physics III*

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# Lucien Hardy, Perimeter Institute

**The causaloid approach to quantum theory and quantum gravity**

March 2008, *Logic, Physics and Quantum Information Theory*

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Quantum theory is probabilistic but has fixed causal structure. General relativity is deterministic but has non-fixed causal structure. It seems that a theory of quantum gravity will inherit the radical features of these two less fundamental theories. Thus, we expect it to be probabilistic and have non-fixed causal structure. In this talk I present a framework, the causaloid formalism, for such theories.

# Chris Heunen, University of Nijmegen

**A topos for algebraic quantum theory**

January 2008, *Categories, Logic and the Foundations of Physics I*

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Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra A induces a topos in which the amalgamation of all its commutative subalgebras comprises a single *commutative* C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum X in the topos, which plays the role of a quantum phase space of the system. States on A become probability integrals on X, and self-adjoint elements of A define functions from X to the pertinent internal real numbers (the interval domain), allowing for a state-proposition pairing. Thus the quantum theory defined by A is turned into a classical theory by restriction to its associated topos.

# Basil Hiley, Birkbeck College London

**Towards a quantum geometry: groupoids, Clifford algebras and shadow manifolds**

May 2008, *Categories, Logic and the Foundations of Physics II*

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I will present some ideas, which although not yet formulated in terms of categories, uses the spirit of category theory especially as articulated by Lawvere. Starting from first principles, I follow Kauffman in making distinctions between two aspects of each indivisible sub-process and then order these processes to form a groupoid, which is then generalised into what I call an algebra of process. The resulting algebra is shown to be isomorphic to a hierarchy of orthogonal Clifford algebras, which, of course, include the Pauli and Dirac algebras. I then exploit the minimal left and right ideals to map from the algebra of process to a vector space inducing a light cone structure thus inverting the usual approach since here the vector space inherits its structure from the fundamental processes. Although I am working with the algebra of process, all of this is still within the conceptual framework used by Clifford himself, namely, classical physics. I then present an argument to generalise this structure to include symplectic Clifford algebras enabling us to introduce the Heisenberg group. Using the techniques applied to orthogonal CAs, I am able to show that the idempotents of this algebra map onto the points of an underlying manifold. Since the symplectic group acts in (x, p) phase space algebra, quantum mechanics demands that this structure is non-commutative and it is this feature that produces shadow manifolds. I will discuss the significance of these results.

# Chris Isham, Imperial College London

**Topos theory in the formulation of theories of physics**

January 2008, *Categories, Logic and the Foundations of Physics I*

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# Peter Johnstone, University of Cambridge

**Topos-theoretic models of the continuum**

May 2008, *Categories, Logic and the Foundations of Physics II*

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We survey a number of different approaches to modelling the continuum in a topos, with particular reference to such questions as whether the real line should be viewed as a space or a locale, the ring-theoretic properties of the continuum, and whether it is sensible to require that all real functions should be continuous or even smooth.

# Lorenzo Maccone, University of Pavia

**A quantum solution to the arrow-of-time dilemma**

July 2008, *Quantum Physics and Logic 2008*

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# Keye Martin, Naval Research Laboratory

**The mechanics of information**

March 2008, *Logic, Physics and Quantum Information Theory*

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**Domain theory and the causal structure of spacetime II**

March 2008, *Clifford Lectures 2008*

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**A domain-theoretic model of qubit channels**

July 2008, *ICALP 2008*

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**How to randomly flip a quantum bit**

July 2008, *Quantum Physics and Logic 2008*

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# Paul-André Melliès, Université Paris Diderot

**Quantum groupoids and logical dualities**

May 2008, *Categories, Logic and the Foundations of Physics II*

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# Prakash Panangaden, McGill University

**The physics of anyons for computer scientists**

March 2008, *Logic, Physics and Quantum Information Theory*

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I give a simple account of the Hall effect and the quantum Hall effect. I give a short account of the spin-statistics theorem and the changes that happen in 2 dimensions which allows for the appearance of anyons.

**Domain theory and the causal structure of spacetime I**

March 2008, *Clifford Lectures 2008*

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# Éric Paquette, McGill University

**Categories and quantum computating with anyons**

March 2008, *Logic, Physics and Quantum Information Theory*

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During this talk I will present the categorical structures necessary to describe anyons; more precisely, the structures that capture the concepts of charges, braids, twist and fusion rules together with the way these structures are used to translate the kinematics of the anyons into the Hilbert space language used to describe quantum computation. We will also present one of the simplest models of non-abelian anyons, the Fibonacci anyons, and explain how to build a set of gates which is universal for quantum computation within this model.

# Simone Perdrix, University of Oxford

**Bases in diagrammatic quantum protocols**

August 2008, *Categories, Logic and the Foundations of Physics III*

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**Partial observation of quantum Turing machine and weaker well-formedness condition**

July 2008, *Quantum Physics and Logic 2008*

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# Mehrnoosh Sadrzadeh, University of Oxford

**What is the vector space content of what we say? … a categorical approach to distributed meaning**

August 2008, *Categories, Logic and the Foundations of Physics III*

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# Peter Selinger, Dalhousie University

**Finite-dimensional Hilbert spaces are complete for dagger compact closed categories**

July 2008, *Quantum Physics and Logic 2008*

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# Rob Spekkens, University of Cambridge

**The power of epistemic restrictions in axiomatizing quantum theory: from trits to qutrits**

July 2008, *Seminar*

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It is common to assert that the discovery of quantum theory overthrew our classical conception of nature. But what, precisely, was overthrown? Providing a rigorous answer to this question is of practical concern, as it helps to identify and analyze quantum technologies that outperform their classical counterparts. It is also of significance for modern physics, where the challenge of applying quantum theory in new realms or moving beyond quantum theory necessitates a deep understanding of the principles upon which it is based. In this talk, I demonstrate that a large part of quantum theory can be obtained from a single innovation relative to classical theories, namely, that there is a fundamental restriction on the sorts of statistical distributions over classical states that can be prepared. This restriction implies a fundamental limit on the amount of knowledge that any observer can have about the classical state. I will consider in particular the case of an arbitrary number of 3-state classical systems, or trits, and I will show that using a particular sort of statistical restriction that appeals to the symplectic structure of the classical state space, one can reproduce the predictions of the stabilizer formalism for qutrits. I will end with a few speculations about the conceptual innovations that might underlie phenomena that can't be derived from a statistical restriction and what might be the origin of the restriction.

# Benoit Valiron, University of Ottawa

**On quantum and probabilistic linear lambda calculi**

July 2008, *Quantum Physics and Logic 2008*

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# Jamie Vicary, University of Oxford

**A categorical framework for the quantum harmonic oscillator**

January 2008, *Categories, Logic and the Foundations of Physics I*

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I will describe a categorical approach to the construction of symmetric Fock space, the state space of the quantum harmonic oscillator. Many of the conventional mathematical tools used to study this system — such as raising and lowering operators, and coherent states — emerge naturally from the category theory, and satisfy the usual equations. However, the formalism is more general than the conventional approach, and I will describe how to construct an infinite variety of 'exotic' Fock spaces. I will finish with the question: "Where has the 'quantumness' come from?"

**Categorical formulation of finite-dimensional C*-algebras**

July 2008, *Quantum Physics and Logic 2008*

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I describe how dagger-Frobenius monoids in the category of finite-dimensional Hilbert spaces are the same as finite-dimensional C*-algebras.

# Steve Vickers, Imperial College London

**Locales via bundles**

August 2008, *Categories, Logic and the Foundations of Physics III*

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Locale theory, a "point-free" approach to topology, can be understood as viewing topological spaces as the spaces of models for logical theories of a particular kind. It has been found effective in a variety of fields, including pure mathematics (deriving from algebraic geometry) and computer science (notably in Abramsky's thesis). One of its most compelling virtues is that it interacts well with constructive mathematics, including the internal mathematics of toposes. In fact, it has a more satisfactory body of constructively valid results than does ordinary point-set topology. Locales have also appeared in the topos approaches to quantum mechanics of Isham and Doering at Imperial and (more explictly) of Heunen, Landsman and Spitters at Nijmegen. Locales have been motivated in a variety of ways, but the path to the constructive virtues can be long and stony. I shall outline a conceptual development in terms of bundles, and going back to a technical result described by Joyal and Tierney in 1984 and known even earlier. Essentially it says that bundles over a space X are equivalent to topological spaces in the internal mathematics of "bundles of sets" (local homeomorphisms) over X. This fits an intuition that "bundle" means a space (the fibre) parametrized by points of the base space. However, there are simple examples to show that this cannot work with a point-set approach to topology - essentially because arbitrary bundles cannot be approximated closely enough by local homeomorphisms. Instead, spaces must be replaced by locales. A key notion is that of "geometric" reasoning, preserved under pullback of bundles, that is more restricted than topos-valid reasoning. It is hoped that the topos-internal reasoning of the Imperial and Nijmegen groups, insofar as it is geometric, can be expressed more intuitively as fibrewise reasoning for bundles.

# Alex Wilce, Susquehanna University

**(No title)**

July 2008, *ICALP 2008*

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# Simon Willerton, University of Sheffield

**Two 2-traces**

August 2008, *Categories, Logic and the Foundations of Physics III*

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In a monoidal category there is a notion of a trace for certain types of endomorphisms — for example the trace of an endomorphism of a finite dimensional vector space. In a monoidal bicategory there are two different notions of trace for endomorphisms which in various cases are 'dual'; I will describe these traces diagrammatically. Several examples from assorted areas will be given and I hope that the audience will come up with some more.

# Andreas Winter, Bristol University

**The mother of all protocols: quantum coding for dummies**

July 2008, *Quantum Physics and Logic 2008*

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# Jon Yard, Caltech

**(No title)**

March 2008, *Logic, Physics and Quantum Information Theory*

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