# Logic, Physics and Quantum Information Theory

## Bellairs Research Centre, March 2008

**(No title)**

Howard Barnum, Los Alamos National Laboratory

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**Measurement-based quantum computation**

Dan Browne, University College London

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**A survey of categorical quantum mechanics**

Bob Coecke, University of Oxford

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

Lucien Hardy, Perimeter Institute

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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.

**The mechanics of information**

Keye Martin, Naval Research Laboratory

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**The physics of anyons for computer scientists**

Prakash Panangaden, McGill University

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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.

**Categories and quantum computating with anyons**

Éric Paquette, McGill University

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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.

**(No title)**

Jon Yard, Caltech

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# Categories, Logic and the Foundations of Physics I

## Imperial College London, January 2008

This was the first workshop in the series *Categories, Logic and the Foundations of Physics*. There seems to be a substantial number of people in the Loxbridge area and beyond with enough interest in these areas to sustain such a series. We also welcome activity from other research strands aiming to gain structural insights into foundational physical theories, for example by means of toy models, operational methodologies for general physical theories, structures for dynamics and space-time etc.

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

Samson Abramsky, University of Oxford

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**Relational topology and quantum gravity**

Louis Crane, Kansas State University

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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.

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

Ross Duncan, University of Oxford

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**A topos for algebraic quantum theory**

Chris Heunen, University of Nijmegen

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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.

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

Chris Isham, Imperial College London

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

Jamie Vicary, University of Oxford

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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?"

# Categories, Logic and the Foundations of Physics II

## Imperial College London, May 2008

The second workshop in the series took place at Imperial College, from 11:00 to 18:45, in the Blackett Laboratory. We are very proud to have heard from six excellent speakers.

**Aspects of duality in 2-categories**

Bruce Bartlett, University of Sheffield

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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’.

**Categories of spectral geometries**

Paolo Bertozzini, Thammasat University

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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.

**Dynamical logic**

Fay Dowker, Imperial College London

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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.

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

Basil Hiley, Birkbeck College London

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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.

**Topos-theoretic models of the continuum**

Peter Johnstone, University of Cambridge

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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.

**Quantum groupoids and logical dualities**

Paul-André Melliès, Université Paris Diderot

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# Categories, Logic and the Foundations of Physics III

## University of Oxford, August 2008

The third workshop on *Categories, Logic and the Foundations of Physics* took place at Oxford University over the weekend of August 23-24. It was requested by the audience at the last workshop to provide tutorials, namely quantum theory for mathematicians and category theory for physicists. At this workshop we indeed provide the first, namely, tutorials on quantum theory for mathematicians. This will include detailed accounts on the many faces of the quantum formalism (von Neumann axiomatics, Dirac notation, Bloch sphere, hints to connections with monoidal categories) and on structural theorems (Gleason's theorem, Kochen-Specken theorem, Bell inequalities, GHZ-correlations, …).

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

John Barrett, University of Nottingham

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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.

**Tutorial: the quantum formalism**

Bob Coecke, University of Oxford

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**Model categories in quantum gravity**

Louis Crane, Kansas State University

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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.

**Tutorial: conceptual issues in quantum theory**

Andreas Doering, Imperial College London

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

Chris Fewster, University of York

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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.

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

Keith Hannabuss, University of Oxford

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**Bases in diagrammatic quantum protocols**

Simone Perdrix, University of Oxford

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

Mehrnoosh Sadrzadeh, University of Oxford

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**Locales via bundles**

Steve Vickers, Imperial College London

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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.

**Two 2-traces**

Simon Willerton, University of Sheffield

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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.

# Clifford Lectures 2008

## Tulane University, March 2008

**Information Flow in Physics, Geometry, Logic and Computation V: A Tale of Dependence and Separation**

Samson Abramsky, University of Oxford

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**Classical versus quantum —- in pictures**

Bob Coecke, University of Oxford

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**The logic of complementary observables**

Ross Duncan, University of Oxford

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

Keye Martin, Naval Research Laboratory

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

Prakash Panangaden, McGill University

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# ICALP 2008

## University of Reykjavik, July 2008

**Interacting quantum observables**

Ross Duncan, University of Oxford

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

Keye Martin, Naval Research Laboratory

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**(No title)**

Alex Wilce, Susquehanna University

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# Seminar

## University of Oxford, July 2008

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

Rob Spekkens, University of Cambridge

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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.

# Quantum Physics and Logic 2008

## University of Reykjavik, July 2008

**On orthomodular posets generated by transition systems**

Luca Bernardinello, University of Milan-Biocca

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

Edward Blakey, University of Oxford

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**Quantum higher types**

Yannick Delbecque, McGill University

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**Classical knowledge for quantum security**

Ellie D'Hondt, Free University of Brussels

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**Toy quantum categories**

Bill Edwards, University of Oxford

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**QML in 15 minutes**

Jonathan Grattage, University of Grenoble

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

Lorenzo Maccone, University of Pavia

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**How to randomly flip a quantum bit**

Keye Martin, Naval Research Laboratory

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

Simone Perdrix, University of Oxford

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

Peter Selinger, Dalhousie University

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**On quantum and probabilistic linear lambda calculi**

Benoit Valiron, University of Ottawa

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

Jamie Vicary, University of Oxford

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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.

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

Andreas Winter, Bristol University

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