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A Categorical Quantum Logic
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2005
"... We define a strongly normalising proofnet calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax ca ..."
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Cited by 22 (5 self)
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We define a strongly normalising proofnet calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.
Delinearizing linearity: projective quantum axiomatics from strong compact closure
 QPL 2005
, 2005
"... ..."
TemperleyLieb Algebra: From Knot Theory to . . .
"... Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics. ..."
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Cited by 11 (2 self)
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Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics.
Kindergarten quantum mechanics — lecture notes
 In: Quantum Theory: Reconsiderations of the Foundations III
, 2005
"... Abstract. These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns ‘doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substanti ..."
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Cited by 9 (7 self)
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Abstract. These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns ‘doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substantial extension of Dirac’s notation, and has a purely algebraic counterpart in terms of socalled Strongly Compact Closed Categories (introduced by Abramsky and I in [3, 4]) which subsumes my Logic of Entanglement [11]. For a survey on the ‘what’, the ‘why ’ and the ‘hows ’ I refer to a previous set of lecture notes [12, 13]. In a last section we provide some pointers to the body of technical literature on the subject.
Abstract physical traces
 THEORY AND APPLICATIONS OF CATEGORIES
, 2005
"... ... in the light of the results in [Abramsky and Coecke LiCS‘04]. The key fact is that the notion of a strongly compact closed category allows abstract notions of adjoint, bipartite projector and inner product to be defined, and their key properties to be proved. In this paper we improve on the defi ..."
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Cited by 6 (5 self)
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... in the light of the results in [Abramsky and Coecke LiCS‘04]. The key fact is that the notion of a strongly compact closed category allows abstract notions of adjoint, bipartite projector and inner product to be defined, and their key properties to be proved. In this paper we improve on the definition of strong compact closure as compared to the one presented in [Abramsky and Coecke LiCS‘04]. This modification enables an elegant characterization of strong compact closure in terms of adjoints and a Yanking axiom, and a better treatment of bipartite projectors.
POVM and Naimark's theorem without sums
"... We introduce an abstract notion of POVM within the categorical quantum mechanical semantics in terms of ..."
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Cited by 6 (4 self)
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We introduce an abstract notion of POVM within the categorical quantum mechanical semantics in terms of
Geometry of Interaction and the Dynamics of Proof Reduction: a tutorial
, 2008
"... Girard’s Geometry of Interaction (GoI) is a program that aims at giving mathematical models of algorithms independently of any extant languages or computing models, thus making it possible to prove general theorems about algorithms. In the context of proof theory, where one views algorithms as proof ..."
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Cited by 5 (2 self)
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Girard’s Geometry of Interaction (GoI) is a program that aims at giving mathematical models of algorithms independently of any extant languages or computing models, thus making it possible to prove general theorems about algorithms. In the context of proof theory, where one views algorithms as proofs and computation as cutelimination, this program translates to providing a mathematical modelling of the dynamics of cutelimination. The kind of logics we deal with, such as Girard’s linear logic, are resource sensitive and have their prooftheory intimately related to various monoidal (tensor) categories. The GoI interpretation of dynamics aims to develop an algebraic/geometric theory of invariants for information flow in networks of proofs.
AN EMBEDDING THEOREM FOR HILBERT CATEGORIES
"... Abstract. We axiomatically define (pre)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small preHilbert category whose monoidal unit is a simple generator embeds (weakly) monoida ..."
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Cited by 5 (5 self)
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Abstract. We axiomatically define (pre)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small preHilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of preHilbert spaces and adjointable maps, preserving adjoint morphisms and all finite (co)limits. An intermediate result that is important in its own right is that the scalars in such a category