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Abstract scalars, loops, and free traced and strongly compact closed categories
 PROCEEDINGS OF CALCO 2005, VOLUME 3629 OF SPRINGER LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of ..."
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Cited by 26 (6 self)
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We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be ‘glued in ’ to the free construction.
Interacting quantum observables
 of Lecture Notes in Computer Science
, 2008
"... Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum d ..."
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Cited by 23 (13 self)
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Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum data encoded in complex phases, and prove a normal form theorem for it. Together these enable us to describe all observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform. All these computations moreover happen within an intuitive diagrammatic calculus. 1
2010), Mathematical foundations for a compositional distributional model of meaning
 Linguistic Analysis (Lambek Festschrift
"... We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the ..."
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Cited by 23 (5 self)
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We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a welltyped sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are ‘lifted ’ to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (welltyped) whole. Importantly, meanings of whole sentences live in a single space, independent of the grammatical structure of the sentence. Hence the innerproduct can be used to compare meanings of arbitrary sentences, as it is for comparing the meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in order to make up the meaning of the whole sentence. A variation of our ‘categorical model ’ which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a Montaguestyle Booleanvalued semantics. 1
Delinearizing linearity: projective quantum axiomatics from strong compact closure
 QPL 2005
, 2005
"... ..."
A Compositional Distributional Model of Meaning
"... We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, namely Lambek’s pregroup semantics. A key observation is that the monoidal category of (finite dimensional) vector spaces, ..."
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Cited by 16 (0 self)
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We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, namely Lambek’s pregroup semantics. A key observation is that the monoidal category of (finite dimensional) vector spaces, linear maps and the tensor product, as well as any pregroup, are examples of compact closed categories. Since, by definition, a pregroup is a compact closed category with trivial morphisms, its compositional content is reflected within the compositional structure of any nondegenerate compact
Introducing categories to the practicing physicist. In: What is Category Theory
 Advanced Studies in Mathematics and Logic 30, pp.45–74, Polimetrica Publishing
, 2006
"... We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra o ..."
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Cited by 12 (7 self)
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We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra of practicing physics. We will not provide rigorous definitions or anything resembling a coherent mathematical theory, but we will take the reader for a journey introducing concepts which are part of category theory in a manner that the physicist will recognize them. 1 Why? Why would a physicist care about category theory, why would he want to know about it, why would he want to show off with it? There could be many reasons. For example, you might find John Baez’s webside one of the coolest in the world. Or you might be fascinated by Chris Isham’s and Lee Smolin’s ideas on the use of topos theory in Quantum Gravity. Also the connections between knot theory, braided categories, and sophisticated mathematical physics such as quantum groups and topological quantum field theory might lure you. Or, if you are also into pure mathematics, you might just appreciate category theory due to its incredible unifying power of mathematical structures and constructions. But there is a far more onthenose reason which is never mentioned. Namely, a category is the exact mathematical structure of practicing physics! What do I mean here by a practicing physics? Consider a physical system of type A (e.g. a qubit, or two qubits, or an electron, or classical measurement data) and perform an operation f on it (e.g. perform a measurement on it) which results in a system possibly of a different type B (e.g. the system together with classical data which encodes the measurement outcome, or, just classical data in the case that the measurement destroyed the system). So typically we have
Types for Quantum Computation
, 2007
"... This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract qua ..."
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Cited by 11 (5 self)
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This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract quantum mechanics. In addition, we demonstrate how these representations may be used to reason about the behaviour of quantum computational processes. Quantum computation is presently mired in lowlevel formalisms, mostly derived directly from matrices over Hilbert spaces. These formalisms are an obstacle to the full understanding and exploitation of quantum effects in informatics since they obscure the essential structure of quantum states and processes. The aim of this work is to introduce higher level tools for quantum mechanics which will be better suited to computation than those presently employed in the field. Inessential details of Hilbert space representations are removed and the informatic structures are presented directly. Entangled states are particularly
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.