{"title":"Dual Objects","authors":"C. Heunen, J. Vicary","doi":"10.1093/oso/9780198739623.003.0003","DOIUrl":"https://doi.org/10.1093/oso/9780198739623.003.0003","url":null,"abstract":"Dual objects are abstract categorical structures that represent the quantum notion of entanglement. We prove a range of important results about dual objects and show how to use them to model quantum teleportation. Dual objects have an important topological representation, in terms of wires bending ‘backwards in time’, and we use this to characterize different sorts of duality structures, including pivotal, ribbon and compact structures. Dual objects interact well with any linear structure available, allowing us to capture linear-algebraic properties such as trace and dimension.","PeriodicalId":314153,"journal":{"name":"Categories for Quantum Theory","volume":"76 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123219174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Basics","authors":"C. Heunen, J. Vicary","doi":"10.1093/oso/9780198739623.003.0009","DOIUrl":"https://doi.org/10.1093/oso/9780198739623.003.0009","url":null,"abstract":"This book assumes familiarity with some basic ideas from category theory, linear algebra and quantum computing. This self-contained chapter gives a quick summary of the essential aspects of these areas, including categories, functors, natural transformations, vector spaces, Hilbert spaces, tensor products, density matrices, measurement and quantum teleportation.","PeriodicalId":314153,"journal":{"name":"Categories for Quantum Theory","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116432607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Monoidal Categories","authors":"Chris Heunen, Jamie Vicary","doi":"10.1093/oso/9780198739623.003.0001","DOIUrl":"https://doi.org/10.1093/oso/9780198739623.003.0001","url":null,"abstract":"A monoidal category is a category equipped with extra data, describing how objects and morphisms can be combined in parallel. This chapter introduces the theory of monoidal categories, including braidings, symmetries and coherence. They form the core of this book, as they provide the basic language with which the rest of the material will be developed. We introduce a visual notation called the graphical calculus, which provides an intuitive and powerful way to work with them. We also introduce the monoidal categories Hilb of Hilbert spaces and linear maps, Set of sets and functions and Rel of sets and relations, which will be used as running examples throughout the book.","PeriodicalId":314153,"journal":{"name":"Categories for Quantum Theory","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126497777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Linear Structure","authors":"C. Heunen, J. Vicary","doi":"10.1093/oso/9780198739623.003.0002","DOIUrl":"https://doi.org/10.1093/oso/9780198739623.003.0002","url":null,"abstract":"Many aspects of linear algebra can be reformulated as categorical structures. This chapter examines abstractions of the base field, zero-dimensional spaces, addition of linear operators, direct sums, matrices, inner products and adjoints. These features are essential for modelling features of quantum theory such as superposition, classical data and measurement.","PeriodicalId":314153,"journal":{"name":"Categories for Quantum Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131282031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complementarity","authors":"C. Heunen, J. Vicary","doi":"10.1093/oso/9780198739623.003.0006","DOIUrl":"https://doi.org/10.1093/oso/9780198739623.003.0006","url":null,"abstract":"Complementarity is a property of a pair of observables being ‘maximally distinct’ from each other and, in this chapter, we analyse this property in categorical terms as a pair of interacting Frobenius structures. Complementary observables play a central role in quantum information theory, and we will see how they can be used to understand the structure of the Deutsch—Jozsa algorithm. We show that complementarity is closely linked to the theory of Hopf algebras. We discuss how many-qubit gates can be modelled using only complementary Frobenius structures, such as controlled negation, controlled phase gates and arbitrary single qubit gates. This leads to the ZX calculus, a sound and complete way to handle quantum computations using only equations in the graphical calculus.","PeriodicalId":314153,"journal":{"name":"Categories for Quantum Theory","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125182289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Frobenius Structures","authors":"C. Heunen, J. Vicary","doi":"10.1093/oso/9780198739623.003.0005","DOIUrl":"https://doi.org/10.1093/oso/9780198739623.003.0005","url":null,"abstract":"A Frobenius structure is a monoid together with a comonoid, which satisfies an interaction law. Frobenius structures have a powerful graphical calculus and we prove a normal form theorem that makes them easy to work with. The Frobenius law itself is justified as a coherence property between daggers and closure of a category. We prove classification theorems for dagger Frobenius structures: in Hilb in terms of operator algebras and in Rel in terms of groupoids. Of special interest is the commutative case—as for Hilbert spaces this corresponds to a choice of basis—and provides a powerful tool to model classical information. We discuss phase gates and the state transfer protocol—as well as modules for Frobenius structures—and show how we can use these to model measurement, controlled operations and quantum teleportation.","PeriodicalId":314153,"journal":{"name":"Categories for Quantum Theory","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123079814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complete Positivity","authors":"C. Heunen, J. Vicary","doi":"10.1093/oso/9780198739623.003.0007","DOIUrl":"https://doi.org/10.1093/oso/9780198739623.003.0007","url":null,"abstract":"Completely positive maps give a notion of quantum process that allows the description of probabilistic effects in quantum theory. We show how to describe this categorically in terms of the CP construction and contrast the behaviour of classical and quantum information in this setting. We also study the behaviour of duality and linear structure under the CP construction.","PeriodicalId":314153,"journal":{"name":"Categories for Quantum Theory","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122817752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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