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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
COMPACTLY ACCESSIBLE CATEGORIES AND QUANTUM KEY DISTRIBUTION
"... Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large n ..."
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Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choiceofduals functor on the compact
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...
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
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.
A quantum double construction in Rel
, 2010
"... We study bialgebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of ..."
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We study bialgebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of
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
unknown title
, 2010
"... We study bialgebras and Hopf algebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras in Rel. In particular, for any group G, we ..."
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We study bialgebras and Hopf algebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras in Rel. In particular, for any group G, we
Appl Categor Struct DOI 10.1007/s1048501092410 Deep Inference and Probabilistic Coherence Spaces
, 2009
"... Abstract This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in ..."
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Abstract This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative selfdual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive