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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 84 (18 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.
Interacting quantum observables: Categorical algebra and diagrammatics
 In Automata, Languages and Programming, ICALP 2008, number 5126 in Lecture Notes in Computer Science
, 2008
"... Abstract: Within an intuitive diagrammatic calculus and corresponding highlevel categorytheoretic algebraic description we axiomatise complementary observables for quantum systems described in finite dimensional Hilbert spaces, and study their interaction. We also axiomatise the phase shifts rela ..."
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Cited by 38 (8 self)
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Abstract: Within an intuitive diagrammatic calculus and corresponding highlevel categorytheoretic algebraic description we axiomatise complementary observables for quantum systems described in finite dimensional Hilbert spaces, and study their interaction. We also axiomatise the phase shifts relative to an observable. The resulting graphical language is expressive enough to denote any quantum physical state of an arbitrary number of qubits, and any processes thereof. The rules for manipulating these result in very concise and straightforward computations with elementary quantum gates, translations between distinct quantum computational models, and simulations of quantum algorithms such as the quantum Fourier transform. They enable the description of the interaction between classical and quantum data in quantum informatic protocols. More specifically, we rely on the previously established fact that in the symmetric monoidal category of Hilbert spaces and linear maps nondegenerate observables correspond to special commutative †Frobenius algebras. This leads to a generalisation of the notion of observable that extends to arbitrary †symmetric monoidal categories (†SMC). We show that any observable in a †SMC comes with an abelian group of phases. We define complementarity of observables in arbitrary †SMCs and prove an elegant diagrammatic characterisation thereof. We show that an important class of complementary observables give rise to a Hopfalgebraic structure, and provide equivalent characterisations thereof. Contents 1. Introduction................................
The Compositional Structure of Multipartite Quantum Entanglement
 IN: PROCEEDINGS OF THE 37TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP), LECTURE NOTES IN COMPUTER SCIENCE
, 2010
"... While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a highlevel, structural understanding of entanglement involving arbitrarily many qubits is a longstanding open problem in quantum computer science. In this paper we expose th ..."
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Cited by 27 (12 self)
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While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a highlevel, structural understanding of entanglement involving arbitrarily many qubits is a longstanding open problem in quantum computer science. In this paper we expose the algebraic and graphical structure of the GHZstate and the Wstate, as well as a purely graphical distinction that characterises the behaviours of these states. In turn, this structure yields a compositional graphical model for expressing general multipartite states. We identify those states, named Frobenius states, which canonically induce an algebraic structure, namely the structure of a commutative Frobenius algebra (CFA). We show that all SLOCCmaximal tripartite qubit states are locally equivalent to Frobenius states. Those that are SLOCCequivalent to the GHZstate induce special commutative Frobenius algebras, while those that are SLOCCequivalent to the Wstate induce what we call antispecial commutative Frobenius algebras. From the SLOCCclassification of tripartite qubit states follows a representation theorem for two dimensional CFAs. Together, a GHZ and a W Frobenius state form the primitives of a graphical calculus. This calculus is expressive enough to generate and reason about arbitrary multipartite states, which are obtained by “composing” the GHZ and Wstates, giving rise to a rich graphical paradigm for general multipartite entanglement.
Complementarity in categorical quantum mechanics
, 2010
"... We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that ( ..."
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Cited by 23 (7 self)
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We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘pointfree’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.
Quantum picturalism
, 2009
"... Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘nocloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to dis ..."
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Cited by 19 (3 self)
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Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘nocloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of ‘quantum teleportation’? We claim that the quantum mechanical formalism doesn’t support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. Using a technical term from computer science, the quantum mechanical formalism is ‘lowlevel’. In this review we present steps towards a diagrammatic ‘highlevel ’ alternative for the Hilbert space formalism, one which appeals to our intuition. The diagrammatic language as it currently stands allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the nocloning theorem, and phenomena such as quantum teleportation. As a logic, it supports ‘automation’: it enables a (classical) computer to reason about interacting quantum systems, prove theorems, and design protocols. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required stepstone towards a deeper conceptual understanding of quantum theory, as well as its
LTL types FRP: Lineartime temporal logic propositions as types, proofs as functional reactive programs
 In Proc. ACM Workshop Programming Languages meets Program Verification
, 2012
"... Functional Reactive Programming (FRP) is a form of reactive programming whose model is pure functions over signals. FRP is often expressed in terms of arrows with loops, which is the type class for a Freyd category (that is a premonoidal category with a cartesian centre) equipped with a premonoid ..."
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Cited by 14 (3 self)
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Functional Reactive Programming (FRP) is a form of reactive programming whose model is pure functions over signals. FRP is often expressed in terms of arrows with loops, which is the type class for a Freyd category (that is a premonoidal category with a cartesian centre) equipped with a premonoidal trace. This type system suffices to define the dataflow structure of a reactive program, but does not express its temporal properties. In this paper, we show that Lineartime Temporal Logic (LTL) is a natural extension of the type system for FRP, which constrains the temporal behaviour of reactive programs. We show that a constructive LTL can be defined in a dependently typed functional language, and that reactive programs form proofs of constructive LTL properties. In particular, implication in LTL gives rise to stateless functions on streams, and the “constrains ” modality gives rise to causal functions. We show that reactive programs form a partially traced monoidal category, and hence can be given as a form of arrows with loops, where the type system enforces that only decoupled functions can be looped.
Breaking Paths in Atomic Flows for Classic Logic
, 2010
"... This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make cruci ..."
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Cited by 12 (5 self)
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This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make crucial use of the path breaker, an atomicflow construction that avoids some nasty termination problems, and that can be used in any proof system with sufficient symmetry. This paper contains an original 2dimensionaldiagram exposition of atomic flows, which helps us to connect atomic flows with other known formalisms.
Finite dimensional Hilbert spaces are complete for dagger compact closed categories
 In the proceedings of QPL 5
, 2008
"... We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces. Keywords: Dagger compact closed categories, Hilbert spaces, completeness. ..."
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Cited by 10 (0 self)
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We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces. Keywords: Dagger compact closed categories, Hilbert spaces, completeness.
Phase groups and the origin of nonlocality for qubits
 Electronic Notes in Theoretical Computer Science
"... We describe a general framework in which we can precisely compare the structures of quantumlike theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operation ..."
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Cited by 9 (1 self)
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We describe a general framework in which we can precisely compare the structures of quantumlike theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operations, and Spekkens’s toy theory. We discover that viewed within our framework these theories are very similar, but differ in one key aspect a four element group we term the phase group which emerges naturally within our framework. In the case of the stabiliser theory this group is Z4 while for Spekkens’s toy theory the group is Z2 × Z2. We further show that the structure of this group is intimately involved in a key physical difference between the theories: whether or not they can be modelled by a local hidden variable theory. This is done by establishing a connection between the phase group, and an abstract notion of GHZ state correlations. We go on to formulate precisely how the stabiliser theory and toy theory are ‘similar ’ by defining a notion of ‘mutually unbiased qubit theory’, noting that all such theories have four element phase groups. Since Z4 and Z2 × Z2 are the only such groups we conclude that the GHZ correlations in this type of theory can only take two forms, exactly those appearing in the stabiliser theory and in Spekkens’s toy theory. The results point at a classification of local/nonlocal behaviours by finite Abelian groups, extending beyond qubits to finitary theories whose observables are all mutually unbiased. 1
Shadows and traces in bicategories
 J. Homotopy Relat. Struct
"... Abstract. Traces in symmetric monoidal categories are wellknown and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative ” traces, s ..."
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Cited by 9 (7 self)
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Abstract. Traces in symmetric monoidal categories are wellknown and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative ” traces, such as the HattoriStallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a “shadow. ” In particular, we prove its functoriality and 2functoriality, which are essential to its applications in fixedpoint theory. Throughout we make use of an appropriate “cylindrical ” type of string