Relating toy models of quantum computation: comprehension, complementarity and dagger mix autonomous categories

Toy models have been used to separate important features of quantum computation from the rich background of the standard Hilbert space model. Category theory, on the other hand, is a general tool to separate components of mathematical structures, and analyze one layer at a time. It seems natural to combine the two approaches, and several authors have already pursued this idea. We explore *categorical comprehension construction* as a tool for adding features to toy models. We use it to comprehend quantum propositions and probabilities within the basic model of finite-dimensional Hilbert spaces. We also analyze complementary quantum observables over the category of sets and relations. This leads into the realm of *test spaces*, a well-studied model. We present one of many possible extensions of this model, enabled by the comprehension construction. Conspicuously, all models obtained in this way carry the same categorical structure, *extending* the familiar dagger compact framework with the complementation operations. We call the obtained structure *dagger mix autonomous*, because it extends mix autonomous categories, popular in computer science, in a similar way like dagger compact structure extends compact categories. Dagger mix autonomous categories seem to arise quite naturally in quantum computation, as soon as complementarity is viewed as a part of the global structure.Comment: 21 pages, 6 figures; Proceedings of Quantum Physics and Logic, Oxford 8-9 April 200

Fibred Coalgebraic Logic and Quantum Protocols

Motivated by applications in modelling quantum systems using coalgebraic techniques, we introduce a fibred coalgebraic logic. Our approach extends the conventional predicate lifting semantics with additional modalities relating conditions on different fibres. As this fibred setting will typically involve multiple signature functors, the logic incorporates a calculus of modalities enabling the construction of new modalities using various composition operations. We extend the semantics of coalgebraic logic to this setting, and prove that this extension respects behavioural equivalence. We show how properties of the semantics of modalities are preserved under composition operations, and then apply the calculational aspect of our logic to produce an expressive set of modalities for reasoning about quantum systems, building these modalities up from simpler components. We then demonstrate how these modalities can describe some standard quantum protocols. The novel features of our logic are shown to allow for a uniform description of unitary evolution, and support local reasoning such as "Alice's qubit satisfies condition" as is common when discussing quantum protocols.Comment: In Proceedings QPL 2013, arXiv:1412.791

Using the Chu construction for generalizing formal concept analysis

Abstract. The goal of this paper is to show a connection between FCA generalisations and the Chu construction on the category ChuCors, the category of formal contexts and Chu correspondences. All needed categorical properties like categorical product, tensor product and its bifunctor properties are presented and proved. Finally, the second order generalisation of FCA is represented by a category built up in terms of the Chu construction

Aczel-Mendler Bisimulations in a Regular Category

Aczel-Mendler bisimulations are a coalgebraic extension of a variety of computational relations between systems. It is usual to assume that the underlying category satisfies some form of axiom of choice, so that the theory enjoys desirable properties, such as closure under composition. In this paper, we accommodate the definition in a general regular category - which does not necessarily satisfy any form of axiom of choice. We show that this general definition 1) is closed under composition without using the axiom of choice, 2) coincides with other types of coalgebraic formulations under milder conditions, 3) coincides with the usual definition when the category has the regular axiom of choice. We then develop the particular case of toposes, where the formulation becomes nicer thanks to the power-object monad, and extend the formalism to simulations. Finally, we describe several examples in Stone spaces, toposes for name-passing, and modules over a ring

Retracing some paths in categorical semantics: From process-propositions-as-types to categorified reals and computers

The logical parallelism of propositional connectives and type constructors extends beyond the static realm of predicates, to the dynamic realm of processes. Understanding the logical parallelism of process propositions and dynamic types was one of the central problems of the semantics of computation, albeit not always clear or explicit. It sprung into clarity through the early work of Samson Abramsky, where the central ideas of denotational semantics and process calculus were brought together and analyzed by categorical tools, e.g. in the structure of interaction categories. While some logical structures borne of dynamics of computation immediately started to emerge, others had to wait, be it because the underlying logical principles (mainly those arising from coinduction) were not yet sufficiently well-understood, or simply because the research community was more interested in other semantical tasks. Looking back, it seems that the process logic uncovered by those early semantical efforts might still be starting to emerge and that the vast field of results that have been obtained in the meantime might be a valley on a tip of an iceberg. In the present paper, I try to provide a logical overview of the gamut of interaction categories and to distinguish those that model computation from those that capture processes in general. The main coinductive constructions turn out to be of this latter kind, as illustrated towards the end of the paper by a compact category of all real numbers as processes, computable and uncomputable, with polarized bisimulations as morphisms. The addition of the reals arises as the biproduct, real vector spaces are the enriched bicompletions, and linear algebra arises from the enriched kan extensions. At the final step, I sketch a structure that characterizes the computable fragment of categorical semantics.Comment: 63 pages, 40 figures; cut two words from the title, tried to improve (without lengthening) Sec.8; rewrote a proof in the Appendi

Sparse deformations of determinant expansions and extensionality of set functions

We introduce an algebraic criterion for the reduction of Z^d-valued maps of subsets to corresponding Z^d-valued maps of elements of a given set. The algebraic structure is given by the expansion of the determinant of two matrices, one of which is generic, where each term of the expansion is deformed through a monomial factor. It is proved that, in broad generality, the deformation is integrable, namely, it is induced by the action of a diagonal matrix on the rows of the generic matrix. This framework allows to extend previous results in the study of signed exponential sums and their applications in statistical physics (statistical amoebas of partition functions), tropical geometry, and integrable systems (soliton solutions of the Kadomtsev-Petviashvili II equation). Furthermore, the hypotheses entailing the integrability of these deformations provide new connections between sparse polynomials, element-set relations, and hyperdeterminants and their factorisation over the ring of Laurent polynomials.Comment: 40 page