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Dagger categories and formal distributions
"... Summary. Monoidal dagger categories play a central role in the abstract quantum mechanics of Abramsky and Coecke. The authors show that a great deal of elementary quantum mechanics can be carried out in these categories; for example, the Born rule emerges naturally. In this paper, we construct a cat ..."
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Summary. Monoidal dagger categories play a central role in the abstract quantum mechanics of Abramsky and Coecke. The authors show that a great deal of elementary quantum mechanics can be carried out in these categories; for example, the Born rule emerges naturally. In this paper, we construct a category of tame formal distributions with coefficients in a commutative associative algebra and show that it is a dagger category. This gives access to a broad new class of models, with the abstract scalars in the sense of Abramsky being the elements of the algebra. We will also consider a subcategory of local formal distributions, based on the ideas of Kac. Locality has been of fundamental significance in various formulations of quantum field theory. Thus our work may provide the possibility of extending the abstract framework to QFT. We also show that these categories of formal distributions are monoidal and contain a nuclear ideal, a weak form of adjunction appropriate for analyzing categories
Pictures of complete positivity in arbitrary dimension
 In Quantum Programming Languages, Electronic Proceedings in Theoretical Computer Science
, 2011
"... Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CPconstruction, that turns any dagger compact category into one with completely positive maps, to arbitrary dimension. Second, we axiomatize when a given category is the result of this constructio ..."
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Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CPconstruction, that turns any dagger compact category into one with completely positive maps, to arbitrary dimension. Second, we axiomatize when a given category is the result of this construction. 1
HigherOrder and Reflexive Action Calculi: Their Type Theory and Models
, 1998
"... Action calculi have been introduced by Milner as a framework for representing models of interactive behaviour. Two natural extensions of action calculi have been proposed: the higherorder action calculi, which add higherorder features to the basic setting, and the reflexive action calculi, which a ..."
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Action calculi have been introduced by Milner as a framework for representing models of interactive behaviour. Two natural extensions of action calculi have been proposed: the higherorder action calculi, which add higherorder features to the basic setting, and the reflexive action calculi, which allow circular bindings of processes and gives recursion in the presense of higherorder features. In this paper, we present simple type theories for action calculi, higherorder action calculi and reflexive action calculi. We also give the categorical models of the extensions, by enriching Power's models of action calculi. As applications, we give a semantic proof of the conservativity of higherorder action calculi over action calculi, and a precise connection with Moggi's computational lambda calculus and notions of computation. We also relate the models of higherorder reflexive action calculi to models of recursive computation, by following the observation that the trace operator of Joya...
Deep Inference and Probabilistic Coherence Spaces
, 2009
"... This paper proposes a definition of categorical model of the deep inference system BV, introduced by Guglielmi. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive category, possibly with negation, with an additional tensor pr ..."
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This paper proposes a definition of categorical model of the deep inference system BV, introduced by Guglielmi. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive category, possibly with negation, with an additional tensor product which, when viewed as a bivariant functor, is linear with a degeneracy condition. We show that this simple definition implies all of the key isomorphisms of the theory. We show that coherence spaces, with Retoré’s noncommutative tensor, is a model.We then consider Girard’s category of probabilistic coherence spaces and show that it contains a selfdual monoidal structure in addition to the ∗autonomous structure exhibited by Girard. This
Operational Theories and Categorical Quantum Mechanics
, 2013
"... A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative ..."
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A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories, and categorical quantum mechanics. The aim of the present paper is to establish strong connections between these two formalisms. We show how models of categorical quantum mechanics have representations as operational theories. We then show how nonlocality can be formulated at this level of generality, and study a number of examples from this point of view, including Hilbert spaces, sets and relations, and stochastic maps. The local, quantum, and nosignalling models are characterized in these terms.
ON THE FUNCTOR ℓ 2
"... and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous lin ..."
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and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous linear maps between Hilbert spaces. 1.
Generalised ProofNets for Compact Categories with Biproducts
, 2009
"... Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic. We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a syste ..."
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Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic. We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a system of proofnets. This logic captures much of the necessary structure needed to represent quantum processes under classical control, while remaining agnostic to the fine details. We demonstrate how to represent quantum processes as proofnets, and show that the dynamic behaviour of a quantum process is captured by the cutelimination procedure for the logic. We show that the cut elimination procedure is strongly normalising: that is, that every legal way of simplifying a proofnet leads to the same, unique, normal form. Finally, taking some initial set of operations
Pipes and Filters: Modelling a Software Architecture Through Relations
, 2002
"... A pipeline is a popular architecture which connects computational components/filers) through connectors (pipes) so that computations are performed in a stream like fashion. The data are transported through the pipes between filers, gradually transforming inputs to outputs. This kind of stream proces ..."
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A pipeline is a popular architecture which connects computational components/filers) through connectors (pipes) so that computations are performed in a stream like fashion. The data are transported through the pipes between filers, gradually transforming inputs to outputs. This kind of stream processing has been made popular through UNIX pipes that serially connect independent components for performing a sequence of tasks. We show in this paper how to formalize this architecture in terms of monads, hereby including relational specifications as special cases. The system is given through a directed acyclic graph the nodes of which carry the computational structure by being labelled with morphisms from the monad, and the edges provide the data for these operations. It is shown how fundamental compositional operations like combining pipes and filers, and refining a system by replacing simple parts through more elaborate ones, are supported through this construction.