**Samson Abramsky, Oxford**

Diagonals, self-reference and the edge of consistency: classical and quantum

Diagonal arguments lie at the root of many fundamental phenomena in the foundations of logic and mathematics - and maybe physics too! In a classic application of category theory to foundations from 1969, Lawvere extracted a simple general argument in an abstract setting which lies at the basis of a remarkable range of results. This clearly exposes the role of a diagonal, i.e. a *copying operation*, in all these results.We shall review some of this material, and give a generalized version of Lawvere's result in the setting of monoidal categories. The same ideas can also be used to give `no go' results, showing that various structural features cannot be combined on pain of a collapse of the structure. This is in the same spirit as the categorical version of No-Cloning and similar results recently obtained by the author. In Computer Science, these show the inconsistency of structures used to model recursion and other computational phenomena with some basic forms of logical structure.In the quantum realm, because of the absence of diagonals (cloning), the situation is less clear-cut. We shall show that various kind of reflexive or self-referential structures do arise in the setting of (infinite dimensional) Hilbert space.

**Howard Barnum, CCS-3: Information Sciences, Los Alamos National Laboratory**

Combining Convex and Categorical Frameworks for Information Processingd Physics

The advent of quantum computation and quantum information science has been accompanied by a revival of the project of characterizing quantum and classical theory within frameworks significantly more general than both. A new twist in the current wave of research is an interest in characterizing theories in terms of their information processing capabilities and properties. Two popular frameworks for recent work are category theory and convex operational models. In this talk, I'll argue for combining the two frameworks and, drawing on joint work with Alex Wilce, describe several ways in which a theory can be described as a category whose objects are convex operational models and whose morphisms are positive maps, possibly with additional structure such as a monoidal product describing the composition of subystems. I'll define notions of saturated and monoidally saturated categories of this type, and argue that these are operationally important as well as mathematically significant notions. For example, quantum theory is monoidally saturated. Connections to joint work with Barrett, Leifer, and Wilce in the convex framework will be explored. I will argue that the combined convexity/categorical framework, and classes of categories abstracted from it, are a good arena for better understanding what is special about the strongly compact (a.k.a. dagger compact) closed categories of Abramsky and Coecke and of Selinger, and thereby better understanding part of what is special about quantum theory.

**Richard Blute**, **Department of Mathematics and Statistics, University of Ottawa**

Categorical Structures in Algebraic Quantum Field Theory

We explore the categorical structures underlying AQFT, and consider possible extensions. Our goal is to take into account ideas suggested by Abramsky and Coecke's recent notion of abstract quantum mechanics.

**Bob Coecke, Perimeter Institute**

Complementarity as a Resource

Andreas Doering, **Theoretical Physics Group, Imperial College**

Why topos theory in the foundations of physics?

An outline of the topos approach to the formulation of physical theories is given, with some emphasis on conceptual and structural aspects. In particular, it will be shown that the topos approach casts some doubt on old dogmas like (a) there is no state space for a quantum system, (b) quantum theory is fundamentally probabilistic, and (c) there is no realist formulation of quantum theory.

**Ross Duncan, Oxford**

Phase Groups and Complementarity

The notion of phase group---a transformation of the state space leaving some observable fixed---arises quite naturally in a variety of quantum-like theories. I will introduce the notion of phase group, and present various examples. In particular I will describe the interaction between phase groups corresponding to complementary pairs of observables.

**Bill Edwards, Oxford University**

The group theoretic origin of non-locality for qubits (joint work with B. Coecke and R. Spekkens)

We describe a categorical framework in which we can precisely compare the structures of quantum-like theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operations, and a toy theory proposed by Spekkens. We discover that viewed within our framework these theories are very similar, but differ in one key aspect - a four element group we term the phase group which emerges naturally within our framework. We further show that the structure of this group is intimately involved in a key physical difference between the theories: whether or not they can be modelled by a local hidden variable theory. We go on to formulate precisely how the stabiliser theory and toy theory are `similar' by defining a notion of `mutually unbiased qubit theory', and go on to show that the GHZ-type correlations in this type of theory can only take two forms, exactly those appearing in the stabiliser theory and those appearing in Spekkens's theory. The results point at a classification of local/non-local behaviours by finite Abelian groups, extending beyond qubits to any finitary theory whose observables are all mutually unbiased.

**Lucien Hardy, Perimeter Institute**

Operational structures as a foundation for probabilistic theories

Work on formulating general probabilistic theories in an operational context has tended to concentrate on the probabilistic aspects (convex cones and so on) while remaining relatively naive about how the operational structure is built up (combining operations to form composite systems, and so on). In particular, an unsophisticated notion of a background time is usually taken for granted. It pays to be more careful about these matters for two reasons. First, by getting the foundations of the operational structure correct it can be easier to prove theorems. And second, if we want to construct new theories (such as a theory of Quantum Gravity) we need to start out with a sufficiently general operational structure before we introduce probabilities. I will present an operational structure which is sufficient to provide a foundation for the probabilistic concepts necessary to formulate quantum theory. According to Bob Coecke, this operational structure corresponds to a symmetric monoidal category. I will then discuss a more general operational framework (which I call Object Oriented Operationalism) which provides a foundation for a more general probabilistic framework which may be sufficient to formulate a theory of Quantum Gravity. This more general operational structure does not admit an obvious category theoretic formulation.

**Ivan T. Ivanov, Vanier College**

Linear Logic for Local Algebras of Observables

Elementary operations on von Neumann algebras allow for interpretation of the Multiplicative-Additive fragment of Linear Logic. Algebras are formulas and states represent their proofs. Cut-elimination, i.e. composition of proofs is composition in the state space of group (or groupoid) von Neumann algebras. The logic connectives have natural interpretations when the operator algebras are algebras of local observables.

**Klaas Landsman, Radboud Universiteit Nijmegen**

The Conway-Kochen-Specker Theorems

Over the past decade, there has been renewed interest in the Kochen-Specker Theorem from 1967, stemming from attempts to prove the existence of free will from it (Conway-Kochen), "nullify" it (Pitowsky, Meyer, Clifton, Kent), generalize it to von Neumann algebras (Doering), or reformulate it in topos theory (Butterfield-Isham and Heunen-Landsman-Spitters).This talk is a survey of these developments and their interrelations.

**Daniel Lehmann, Hebrew University of Jerusalem**

Towards an abstract description of tensor product

Tensor product is described in a family of categories that includes Set and Hilbert spaces. Such categories admit a "scalar" object which enables a definition of bi-arrows with two domains, generalizing functions of two variables. The tensor product is characterized by the expected universal property relating bi-arrows to arrows.

**Prakash Panangaden, McGill University**

Discrete Quantum Causal Dynamics

We give a mathematical framework to describe the evolution of quantum systems subject to finitely many interactions with classical apparatuus and with each other. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently, but may also interact. The evolution is coded in a mathematical structure in such a way that the properties of causality, covariance and entanglement are faithfully represented. The key to this scheme is to use a special family of spacelike slices -- we call them locative -- that are not so large as to permit acausal influences but large enough to capture nonlocal correlations. I will briefly describe how the dynamics can be described as a functor to a suitable category of Hilbert spaces and will also give some connections with logic.

**Marni Dee Sheppeard, Oxford University**

A little categorified arithmetic from entanglement

In topos quantum gravity, one would like to consider categorical axioms for quantum mechanics based solely on complementary observables and not on monoidal categories of Hilbert spaces. An example is the characterisation of arithmetic in a topos, and its generalisation to quantum systems. We discuss elementary finite field arithmetic in this context, with some inspiration from the classification of entanglement for qubits.

**Rafael Sorkin, Perimeter Institute**

What is a quantal reality?

I will rephrase the question, "What is a quantal reality?" as "What is a quantal history?" (the word history having here the same meaning as in the phrase sum-over-histories). The answer I will propose modifies the rules of logical inference in order to resolve a contradiction between the idea of reality as a single history and the principle that events of zero measure cannot happen (the Kochen-Specker paradox being a classic expression of this contradiction). The so-called measurement problem is then solved if macroscopic events satisfy classical logic, and this can in principle be decided by a calculation. The resulting conception of reality involves neither multiple worlds nor external observers. It is therefore suitable for quantum gravity in general and causal sets in particular.

**Robert Spekkens, Perimeter Institute**

Quantum analogues of Bayes' theorem, sufficient statistics and the pooling problem

The notion of a conditional probability is critical for Bayesian reasoning. Bayes’ theorem, the engine of inference, concerns the inversion of conditional probabilities. Also critical are the concepts of conditional independence and sufficient statistics. The conditional density operator introduced by Leifer is a natural generalization of conditional probability to quantum theory. This talk will pursue this generalization to define quantum analogues of Bayes' theorem, conditional independence and sufficient statistics. These can be used to provide simple proofs of certain well-known results in quantum information theory, such as the isomorphism between POVMs and convex decompositions of a mixed state and the remote collapse postulate, and to prove some novel results on how to pool quantum states. This is joint work with Matt Leifer. I will also briefly discuss the possibility of a diagrammatic calculus for classical and quantum Bayesian inference (joint work with Bob Coecke).

**Mike Stay**, **Google**

Physics, Topology, Logic, and Computation: A Rosetta Stone

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository talk, I'll make some of these analogies precise using the concept of "closed braided monoidal category.

**Jamie Vicary, Oxford**

Higher-dimensional quantum mechanics

We sketch some ideas about how higher-dimensional categories could be used to extend conventional quantum mechanics. The physical motivation comes from quantum field theory, for which higher-dimensional category theory is very relevant. We discuss how this new approach would affect familiar aspects of quantum theory, such as observables and the Copenhagen interpretation. Few solid answers will be given, but hopefully some discussion will be generated!