A Survey of Graphical Languages for Monoidal Categories
TL;DR: In this article, a reference guide to various notions of monoidal categories and their associated string diagrams is presented, which is useful not only to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning.
Abstract: This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study Where possible, we provide pointers to more rigorous treatments in the literature Where we include results that have only been proved in special cases, we indicate this in the form of caveats
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TL;DR: In this article, it was shown that the purification principle is equivalent to the existence of a reversible realization of every physical process, that is, to the fact that a physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded.
Abstract: We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, that is, to the fact that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamiolkowski isomorphism in quantum theory. Such an isomorphism allows one to prove most of the basic features of quantum theory, like, e.g., existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.
512 citations
31 Mar 2017
TL;DR: This entirely diagrammatic presentation of quantum theory represents the culmination of ten years of research, uniting classical techniques in linear algebra and Hilbert spaces with cutting-edge developments in quantum computation and foundations.
Abstract: The unique features of the quantum world are explained in this book through the language of diagrams, setting out an innovative visual method for presenting complex theories. Requiring only basic mathematical literacy, this book employs a unique formalism that builds an intuitive understanding of quantum features while eliminating the need for complex calculations. This entirely diagrammatic presentation of quantum theory represents the culmination of ten years of research, uniting classical techniques in linear algebra and Hilbert spaces with cutting-edge developments in quantum computation and foundations. Written in an entertaining and user-friendly style and including more than one hundred exercises, this book is an ideal first course in quantum theory, foundations, and computation for students from undergraduate to PhD level, as well as an opportunity for researchers from a broad range of fields, from physics to biology, linguistics, and cognitive science, to discover a new set of tools for studying processes and interaction.
360 citations
TL;DR: The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework.
Abstract: This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z- and X-spin observables, which yields a scaled variant of a bialgebra.
353 citations
Cites background or methods from "A Survey of Graphical Languages for..."
...Moreover, when expressed in the graphical language, the coherence conditions for SMCs become trivial as a consequence of some very powerful theorems, so they play no further role in this paper....
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...These two are related by the fact that New Journal of Physics 13 (2011) 043016 (http://www.njp.org/) 4 there is a tight correspondence between graphical languages and SMCs [47, 69], tracing back to Penrose’s work on tensor networks [61]....
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...We studied its mathematical underpinning in great detail, in particular: • We obtained a purely diagrammatic characterization of complementarity that extends to observable structures in arbitrary †-SMCs, in terms of the Hopf law: • We identified a strong form of complementarity for observable structures in arbitrary †-SMCs when the observable structures form a scaled bialgebra: We identified a number of equivalent alternative formulations: k k k k kk k ‘ k ‘ k ‘ k k , k k , k k , • We identified a group structure on phases for observable structures in arbitrary †-SMCs, and proved a generalization of the spider rules, now involving phases: ....
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...Observable structures with coinciding †-compact structures First we define complementarity for observable structures in arbitrary †-SMCs in a manner that makes explicit reference to their classical points, simply in analogy to the usual definition in the Hilbert space quantum theory, and then we show that this definition can be equivalently restated without any reference to points....
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...However, the axioms of SMCs are rather weak, so the isomorphism principle will not suffice....
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TL;DR: A general mathematical definition of resource theory is proposed and general theorems about how resource theories can be constructed from theories of processes with a subclass of processes that are freely implementable are proved.
Abstract: Many fields of science investigate states and processes as resources. Chemistry, thermodynamics, Shannon's theory of communication channels, and the theory of quantum entanglement are prominent examples. Questions addressed by these theories include: Which resources can be converted into which others? At what rate can many copies of one resource be converted into many copies of another? Can a catalyst enable a conversion? How to quantify a resource? We propose a general mathematical definition of resource theory. We prove general theorems about how resource theories can be constructed from theories of processes with a subclass of processes that are freely implementable. These define the means by which costly states and processes can be interconverted. We outline how various existing resource theories fit into our framework, which is a first step in a project of identifying universal features and principles of resource theories. We develop a few general results concerning resource convertibility.
334 citations
Cites background from "A Survey of Graphical Languages for..."
...In fact, there exist powerful theorems which establish that equational reasoning within an SMC is in one-to-one correspondence with deformation of diagrams [32, 44]....
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TL;DR: The ZX-calculus as mentioned in this paper is an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information.
Abstract: This paper has two tightly intertwined aims: (i) To introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatise complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework.
Using the well-studied canonical correspondence between graphical calculi and symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z and X spin observables, which yields a scaled variant of a bialgebra.
314 citations
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01 Jan 1971
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
9,254 citations
"A Survey of Graphical Languages for..." refers background in this paper
...ssed in Section 5. 8 3.1 (Planar) monoidal categories A monoidal category (also sometimes called tensor category) is a category with an associative unital tensor product. More specifically: Definition ([29, 23]). A monoidal category is a category with the following additional structure: • a new operation A ⊗B on objects and a new object constant I; • a new operation on morphisms: if f : A → C and g : B → D,...
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...rlier draft. 4 2 Categories We only give the most basic definitions of categories, functo rs, and natural transformations. For a gentler introduction, with more details and examples, see e.g. Mac Lane [29]. Definition. A category Cconsists of: • a class |C| of objects, denoted A, B, C, ...; • for each pair of objects A,B, a set homC(A,B) of morphisms, which are denoted f : A → B; • identity morphisms id...
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TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.
Abstract: Linear logic was introduced by Girard in 1987 [11] . Since then many results have supported Girard' s statement, \"Linear logic is a resource conscious logic,\" and related slogans . Increasingly, computer scientists have come to recognize linear logic as an expressive and powerful logic with connection s to a variety of topics in computer science . This column presents a.n intuitive overview of linear logic, some recent theoretical results, an d summarizes several applications of linear logic to computer science . Other introductions to linear logic may be found in [12, 361 .
2,304 citations
"A Survey of Graphical Languages for..." refers background or methods in this paper
...The resulting theory is called the theory of proof nets, and was first given by Girard for unit-free multiplicative linear logic [17]....
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...Both of these notions are models of multiplicative linear logic [17]....
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TL;DR: In this paper, a systematic treatment of topological quantum field theories (TQFT's) in 3D is presented, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups.
Abstract: This monograph provides a systematic treatment of topological quantum field theories (TQFT's) in three dimensions, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups. The author, one of the leading experts in the subject, gives a rigorous and self-contained exposition of new fundamental algebraic and topological concepts that emerged in this theory. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFT's and 2-dimensional modular functors from so-called modular categories. This gives new knot and 3-manifold invariants as well as linear representations of the mapping class groups of surfaces. In Part II the machinery of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFT's constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and Kauffman's skein modules. This book is accessible to graduate students in mathematics and physics with a knowledge of basic algebra and topology. It will be an indispensable source for everyone who wishes to enter the forefront of this rapidly growing and fascinating area at the borderline of mathematics and physics. Most of the results and techniques presented here appear in book form for the first time.
1,506 citations
TL;DR: This contribution was made possible only by the miraculous fact that the first members of the Editorial Board were sharing the same conviction about the necessity of Theoretical Computer Science.
1,480 citations