84 research outputs found

    On weak Hopf symmetry and weak Hopf quantum double model

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    Symmetry is a central concept for classical and quantum field theory, usually, symmetry is described by a finite group or Lie group. In this work, we introduce the weak Hopf algebra extension of symmetry, which arises naturally in anyonic quantum systems; and we establish weak Hopf symmetry breaking theory based on the fusion closed set of anyons. As a concrete example, we implement a thorough investigation of the quantum double model based on a given weak Hopf algebra and show that the vacuum sector of the model has weak Hopf symmetry. The topological excitations and ribbon operators are discussed in detail. The gapped boundary and domain wall theories are also established, we show that the gapped boundary is algebraically determined by a comodule algebra, or equivalently, a module algebra; and the gapped domain wall is determined by the bicomodule algebra, or equivalently, a bimodule algebra. The microscopic lattice constructions of the gapped boundary and domain wall are discussed in detail. We also introduce the weak Hopf tensor network states, via which we solve the weak Hopf quantum double lattice models on closed and open surfaces. The duality of the quantum double phases is discussed in the last part.Comment: 63 pages. Comments welcom

    Algebraic Structure of Topological and Conformal Field Theories

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    Quantum field theories (QFTs) are geometric and analytic in nature. With enough symmetry, some QFTs may admit partial or fully algebraic descriptions. Topological and conformal field theories are prime examples of such QFTs. In this thesis, the algebraic structure of 2+1D Topological Quantum Field Theories (TQFTs) and associated Conformal Field Theories (CFTs) is studied. The line operators of 2 + 1D TQFTs and their correlation functions are captured by an algebraic structure called a Modular Tensor Category (MTC). A basic property of line operators is their operator product expansion. This is captured by the fusion rules of the MTC. We study the existence and consequences of special fusion rules where two line operators fuse to give a unique outcome. There is a natural action of a Galois group on MTCs which allows us to jump between points in the space of TQFTs. We study how the physical properties of a TQFT like its symmetries and gapped boundaries transform under Galois action. We also study how Galois action interacts with other algebraic operations on the space of TQFTs like gauging and anyon condensation. Moreover, we show that TQFTs which are invariant under Galois action are very special. Such Galois invariant TQFTs can be constructed from gauging symmetries of certain simple abelian TQFTs. TQFTs also admit gapless boundaries. In particular, 1+1D Rational CFTs (RCFTs) and 2+1D TQFTs are closely related. Given a chiral algebra, the consistent partition functions of an RCFT are classified by surface operators in the bulk 2 + 1D TQFT. On the other hand, Narain RCFTs can be constructed from quantum error-correcting codes (QECCs). We give a general map from Narain RCFTs to QECCs. We explore the role of topological line operators of the RCFT in this construction and use this map to give a quantum code theoretic interpretation of orbifolding

    Categorical Modelling of Logic Programming: Coalgebra, Functorial Semantics, String Diagrams

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    Logic programming (LP) is driven by the idea that logic subsumes computation. Over the past 50 years, along with the emergence of numerous logic systems, LP has also grown into a large family, the members of which are designed to deal with various computation scenarios. Among them, we focus on two of the most influential quantitative variants are probabilistic logic programming (PLP) and weighted logic programming (WLP). In this thesis, we investigate a uniform understanding of logic programming and its quan- titative variants from the perspective of category theory. In particular, we explore both a coalgebraic and an algebraic understanding of LP, PLP and WLP. On the coalgebraic side, we propose a goal-directed strategy for calculating the probabilities and weights of atoms in PLP and WLP programs, respectively. We then develop a coalgebraic semantics for PLP and WLP, built on existing coalgebraic semantics for LP. By choosing the appropriate functors representing probabilistic and weighted computation, such coalgeraic semantics characterise exactly the goal-directed behaviour of PLP and WLP programs. On the algebraic side, we define a functorial semantics of LP, PLP, and WLP, such that they three share the same syntactic categories of string diagrams, and differ regarding to the semantic categories according to their data/computation type. This allows for a uniform diagrammatic expression for certain semantic constructs. Moreover, based on similar approaches to Bayesian networks, this provides a framework to formalise the connection between PLP and Bayesian networks. Furthermore, we prove a sound and complete aximatization of the semantic category for LP, in terms of string diagrams. Together with the diagrammatic presentation of the fixed point semantics, one obtain a decidable calculus for proving the equivalence between propositional definite logic programs

    String Diagram Rewrite Theory I: Rewriting with Frobenius Structure

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    String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses quite a few additional challenges. This work systematises and expands a series of recent conference papers, laying down such a foundation. As a first step, we focus on the case of rewrite systems for string diagrammatic theories that feature a Frobenius algebra. This common structure provides a more permissive notion of composition than the usual one available in monoidal categories, and has found many applications in areas such as concurrency, quantum theory, and electrical circuits. Notably, this structure provides an exact correspondence between the syntactic notion of string diagrams modulo Frobenius structure and the combinatorial structure of hypergraphs. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting. We prove this interpretation to be sound and complete and we also show that the approach can be generalised to rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs

    CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond

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    These notes are the output of a decade of research on how the results about dilations of one-parameter CP-semigroups with the help of product systems, can be put forward to d-parameter semigroups - and beyond. While exisiting work on the two- and d-parameter case is based on the approach via the Arveson-Stinespring correspondence of a CP-map by Muhly and Solel (and limited to von Neumann algebras), here we explore consequently the approach via Paschke's GNS-correspondence of a CP-map by Bhat and Skeide. (A comparison is postponed to Appendix A(iv).) The generalizations are multi-fold, the difficulties often enormous. In fact, our only true if-and-only-if theorem, is the following: A Markov semigroup over (the opposite of) an Ore monoid admits a full (strict or normal) dilation if and only if its GNS-subproduct system embeds into a product system. Already earlier, it has been observed that the GNS- (respectively, the Arveson-Stinespring) correspondences form a subproduct system, and that the main difficulty is to embed that into a product system. Here we add, that every dilation comes along with a superproduct system (a product system if the dilation is full). The latter may or may not contain the GNS-subproduct system; it does, if the dilation is strong - but not only. Apart from the many positive results pushing forward the theory to large extent, we provide plenty of counter examples for almost every desirable statement we could not prove. Still, a small number of open problems remains. The most prominent: Does there exist a CP-semigroup that admits a dilation, but no strong dilation? Another one: Does there exist a Markov semigroup that admits a (necessarily strong) dilation, but no full dilation?Comment: (SEE ALSO COMMENTS TO V2.) In this revision: Corrected a bad missprint caused by a cut-and-paste error during index production, which made almost unreadable a prerequisite for one of our major results. (Last paragraph p.66 (after "dual" and Corollary 7.7.) Further minor corrections to the inde

    String Diagram Rewrite Theory I: Rewriting with Frobenius Structure

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    String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology.In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses quite a few additional challenges.This work systematises and expands a series of recent conference papers, laying down such a foundation. As a first step, we focus on the case of rewrite systems for string diagrammatic theories that feature a Frobenius algebra. This common structure provides a more permissive notion of composition than the usual one available in monoidal categories, and has found many applications in areas such as concurrency, quantum theory, and electrical circuits. Notably, this structure provides an exact correspondence between the syntactic notion of string diagrams modulo Frobenius structure and the combinatorial structure of hypergraphs.Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting. We prove this interpretation to be sound and complete and we also show that the approach can be generalised to rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs

    String diagram rewrite theory II: Rewriting with symmetric monoidal structure

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    Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras

    Exponential Modalities and Complementarity (extended abstract)

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    The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics. The paper uses a formulation of quantum systems based on dagger-linear logic, whose categorical semantics lies in mixed unitary categories, and a formulation of measurement therein. The main result exhibits a complementary system as the result of measurements on free exponential modalities. Recalling that, in linear logic, exponential modalities have two distinct but dual components, ! and ?, this shows how these components under measurement become "compacted" into the usual notion of complementary Frobenius algebras from categorical quantum mechanics.Comment: In Proceedings ACT 2021, arXiv:2211.01102. A full version of this paper, containing all proofs, appears at arXiv:2103:0519

    Space and time in monoidal categories

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    The use of categorical methods is becoming more prominent and successful in both physics and computer science. The basic idea is that objects of a category can represent systems, and morphisms can model the processes that transform those systems. We can see parts of computational protocols or physical processes as morphisms, which, when appropriately combined using tensor products and categorical composition, model the protocol or process as a whole. However, in doing so, some information about the protocols or processes is forgotten, namely in what location of spacetime did the events involved take place, and what was the causal structure among them. The goal of this thesis is to explore how these categorical models can be enhanced to include information on the spacetime location and causal structure of events. First, we introduce the theory of subunits, which are subobjects of the monoidal unit for which a canonical isomorphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules, and under mild conditions they endow any monoidal category with a topological intuition. We introduce and study well-behaved notions of restriction, localisation, and support. Subunits in general form only a semilattice, but we develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.  Afterwards, we introduce a number of constructions to explore how the theory of subunits can be used in practice. Inspired by logical clocks, we define a diagrammatic category where we can capture simple protocols and their causal structure. To progress towards more detailed spacetime and causal information, we define the category of protocols, which formalises the idea of letting a morphism from a category be supported in a different category. This allows us to have one category to model the systems and processes and another one to model spacetime. In particular, we can treat both toy models of spacetime and more realistic ones in the same mathematical footing. A notion of causal structure is defined for monoidal categories, and a generalisation of the usual causal analysis in physics for points to arbitrary regions is provided. We give examples of protocols seen as diagrams and as objects in the category of protocols, both with toy models of spacetime as well as with more realistic ones
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