Jamie Vicary

A new description of orthogonal bases

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the †-functor. Moreover, this characterisation can be interpreted operationally, since the †-Frobenius structure allows the cloning and deletion of basis vectors. That is, we capture the basis vectors by relying on their ability to be cloned and deleted. Since this ability distinguishes classical data from quantum data, our result has important implications for categorical quantum mechanics.

Categorical Formulation of Finite-Dimensional Quantum Algebras

We describe how †-Frobenius monoids give the correct categorical description of two kinds of finite-dimensional ‘quantum algebras’. We develop the concept of an involution monoid, and use it to show that finite-dimensional C*-algebras are the same as special unitary †-Frobenius monoids in the category of finite-dimensional complex Hilbert spaces. The spectral theorems for commutative C*-algebras and for normal operators are discussed from this perspective, and we formulate them in an explicitly categorical way. We examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos, and are motivated to define the term finite quantum Boolean topos. We end with a study of the 2-categorical generalisation, and show that 2-H*-algebras are the same as †-Frobenius pseudomonoids in the bicategory of 2-Hilbert spaces.

Higher Semantics of Quantum Protocols

We propose a higher semantics for the description of quantum protocols, which deals with quantum and classical information in a unified way. Central to our approach is the modelling of classical data by information transfer to the environment, and the use of 2-category theory to formalize the resulting framework. This 2-categorical semantics has a graphical calculus, the diagrams of which correspond exactly to physically-implementable quantum procedures. Quantum teleportation in its most general sense is reformulated as the ability to remove correlations between a quantum system and its environment, and is represented by an elegant graphical identity. We use this new formalism to describe two new families of quantum protocols.

A Categorical Framework for the Quantum Harmonic Oscillator

This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an entirely general one in which Hilbert spaces play no special role.Generalised coherent states arise through the hom-set isomorphisms defining the adjunction, and we prove that they are eigenstates of the lowering operators. Generalised exponentials also emerge naturally in this setting, and we demonstrate that coherent states are produced by the exponential of a raising morphism acting on the zero-particle state. Finally, we examine all of these constructions in a suitable category of Hilbert spaces, and find that they reproduce the conventional mathematical structures.

Topological Structure of Quantum Algorithms

We use a categorical topological semantics to examine the Deutsch-Jozsa, hidden subgroup and single-shot Grover algorithms. This reveals important structures hidden by conventional algebraic presentations, and allows novel proofs of correctness via local topological operations, giving for the first time a satisfying high-level explanation for why these procedures work. We also investigate generalizations of these algorithms, providing improved analyses of those already in the literature, and a new generalization of the single-shot Grover algorithm.We use a categorical topological semantics to examine the Deutsch-Jozsa, hidden subgroup and single-shot Grover algorithms. This reveals important structures hidden by conventional algebraic presentations, and allows novel proofs of correctness via local topological operations, giving for the first time a satisfying high-level explanation for why these procedures work. We also investigate generalizations of these algorithms, providing improved analyses of those already in the literature, and a new generalization of the single-shot Grover algorithm. 1 Overview 1.

Completeness of †-categories and the complex numbers

The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this manner satisfies certain completeness properties, then it necessarily includes the complex numbers as a mathematical ingredient. Central to our approach are the techniques of category theory, and we introduce a new category-theoretical tool, called the dagger-limit, which governs the way in which systems can be combined to form larger systems. These dagger-limits can be used to characterize the dagger-functor on the category of finite-dimensional Hilbert spaces, and so can be used as an equivalent definition of the inner product. One of our main results is that in a nontrivial monoidal dagger-category with all finite dagger-limits and a simple tensor unit, the semiring of scalars embeds into an involutive field of characteristic 0 and orderable fixed field.

Globular: An Online Proof Assistant for Higher-Dimensional Rewriting

This article introduces Globular, an online proof assistant for the formalization and verification of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their construction with a point-and- click interface, and performs type checking to prevent incorrect rewrites. Hosted on the web, it has a low barrier to use, and allows hyperlinking of formalized proofs directly from research papers. It allows the formalization of proofs from logic, topology and algebra which are not formalizable by other methods, and we give several examples.

Bicategorical Semantics for Nondeterministic Computation

We present a topological bicategorical syntax for the interaction between public and private information in classical information theory. This allows high-level graphical definitions of encrypted communication and secret sharing, including a characterization of their security properties, which are automatically satisfied with no extra axioms required. This analysis shows that these protocols have an identical abstract form to the quantum teleportation and dense coding procedures, giving a concrete mathematical analogy between quantum and classical computing. Specific implementations of these protocols as nondeterministic classical procedures are recovered by applying our formalism in a symmetric monoidal bicategory of matrices of relations. http://www.elsevier.nl/locate/entcs.

Abstract structure of unitary oracles for quantum algorithms.

We show that a pair of complementary dagger-Frobenius algebras, equipped with a self-conjugate comonoid homomorphism onto one of the algebras, produce a nontrivial unitary morphism on the product of the algebras. This gives an abstract understanding of the structure of an oracle in a quantum computation, and we apply this understanding to develop a new algorithm for the deterministic identification of group homomorphisms into abelian groups. We also discuss an application to the categorical theory of signal-flow networks.

WEDS: a Web services-based environment for distributed simulation

Web services have the potential to radically enhance the ability of researchers to make use of distributed computing resources, but jargon and a plethora of standards make their use almost impossible for the scientist without prior experience of the necessary technologies. A powerful and simple WSRF-based middleware scheme is presented, designed to let scientists remotely deploy single or multiple instances of a pre-existing code across multiple resources, and giving steering, visualization and workflow functionality with only simple modifications to program code. It is hoped that the development and implementation of such a toolkit will be relevant not only to the problem of deploying workstation-class codes in real time, but also the move towards more tractable alternatives to the Globus toolkit for deployment of processes in a high-performance computing environment.