Categorical Quantum Models and Logics

Categorical Quantum Models and Logics PDF Author: Chris Heunen
Publisher: Amsterdam University Press
ISBN: 9085550246
Category : Mathematics
Languages : en
Pages : 214

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Book Description
This dissertation studies the logic behind quantum physics, using category theory as the principal tool and conceptual guide. To do so, principles of quantum mechanics are modeled categorically. These categorical quantum models are justified by an embedding into the category of Hilbert spaces, the traditional formalism of quantum physics. In particular, complex numbers emerge without having been prescribed explicitly. Interpreting logic in such categories results in orthomodular property lattices, and furthermore provides a natural setting to consider quantifiers. Finally, topos theory, incorporating categorical logic in a refined way, lets one study a quantum system as if it were classical, in particular leading to a novel mathematical notion of quantum-

Categorical Quantum Models and Logics

Categorical Quantum Models and Logics PDF Author: Chris Heunen
Publisher: Amsterdam University Press
ISBN: 9085550246
Category : Mathematics
Languages : en
Pages : 214

Get Book Here

Book Description
This dissertation studies the logic behind quantum physics, using category theory as the principal tool and conceptual guide. To do so, principles of quantum mechanics are modeled categorically. These categorical quantum models are justified by an embedding into the category of Hilbert spaces, the traditional formalism of quantum physics. In particular, complex numbers emerge without having been prescribed explicitly. Interpreting logic in such categories results in orthomodular property lattices, and furthermore provides a natural setting to consider quantifiers. Finally, topos theory, incorporating categorical logic in a refined way, lets one study a quantum system as if it were classical, in particular leading to a novel mathematical notion of quantum-

Categories for Quantum Theory

Categories for Quantum Theory PDF Author: Chris Heunen
Publisher: Oxford University Press
ISBN: 0191060062
Category : Mathematics
Languages : en
Pages : 320

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Book Description
Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition, and a conceptual way to understand many high-level quantum phenomena. This text lays the foundation for this categorical quantum mechanics, with an emphasis on the graphical calculus which makes computation intuitive. Biproducts and dual objects are introduced and used to model superposition and entanglement, with quantum teleportation studied abstractly using these structures. Monoids, Frobenius structures and Hopf algebras are described, and it is shown how they can be used to model classical information and complementary observables. The CP construction, a categorical tool to describe probabilistic quantum systems, is also investigated. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Prior knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text links with many other areas are highlighted, such as representation theory, topology, quantum algebra, knot theory, and probability theory, and nonstandard models are presented, such as sets and relations. All results are stated rigorously, and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.

Logic and Algebraic Structures in Quantum Computing

Logic and Algebraic Structures in Quantum Computing PDF Author: Jennifer Chubb
Publisher: Cambridge University Press
ISBN: 110703339X
Category : Computers
Languages : en
Pages : 355

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Book Description
Experts in the field explore the connections across physics, quantum logic, and quantum computing.

Handbook of Quantum Logic and Quantum Structures

Handbook of Quantum Logic and Quantum Structures PDF Author: Kurt Engesser
Publisher: Elsevier
ISBN: 0080931669
Category : Mathematics
Languages : en
Pages : 727

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Book Description
Quantum mechanics is said to be the most successful physical theory ever. It is, in fact, unique in its success when applied to concrete physical problems. On the other hand, however, it raises profound conceptual problems that are equally unprecedented. Quantum logic, the topic of this volume, can be described as an attempt to cast light on the puzzle of quantum mechanics from the point of view of logic. Since its inception in the famous 1936 paper by Birkhoff and von Neumann entitled, "The logic of quantum mechanics, quantum logic has undergone an enormous development. Various schools of thought and approaches have emerged, and there are a variety of technical results. The chapters of this volume constitute a comprehensive presentation of the main schools, approaches and results in the field of quantum logic. - Authored by eminent scholars in the field - Material presented is of recent origin representing the frontier of the subject - Provides the most comprehensive and varied discussion of Quantum Mechanics available

Mathematical Foundations of Information Flow

Mathematical Foundations of Information Flow PDF Author: Samson Abramsky
Publisher: American Mathematical Soc.
ISBN: 0821849239
Category : Mathematics
Languages : en
Pages : 282

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Book Description
This volume is based on the 2008 Clifford Lectures on Information Flow in Physics, Geometry and Logic and Computation, held March 12-15, 2008, at Tulane University in New Orleans, Louisiana. The varying perspectives of the researchers are evident in the topics represented in the volume, including mathematics, computer science, quantum physics and classical and quantum information. A number of the articles address fundamental questions in quantum information and related topics in quantum physics, using abstract categorical and domain-theoretic models for quantum physics to reason about such systems and to model spacetime. Readers can expect to gain added insight into the notion of information flow and how it can be understood in many settings. They also can learn about new approaches to modeling quantum mechanics that provide simpler and more accessible explanations of quantum phenomena, which don't require the arcane aspects of Hilbert spaces and the cumbersome notation of bras and kets.

Computation, Logic, Games, and Quantum Foundations - The Many Facets of Samson Abramsky

Computation, Logic, Games, and Quantum Foundations - The Many Facets of Samson Abramsky PDF Author: Bob Coecke
Publisher: Springer
ISBN: 3642381642
Category : Computers
Languages : en
Pages : 381

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Book Description
This Festschrift volume, published in honor of Samson Abramsky, contains contributions written by some of his colleagues, former students, and friends. In celebration of the 60th birthday of Samson Abramsky, a conference was held in Oxford, UK, during May 28-30, 2010. The papers in this volume represent his manifold contributions to semantics, logic, games, and quantum mechanics.

Uncountably Categorical Theories

Uncountably Categorical Theories PDF Author: Boris Zilber
Publisher: American Mathematical Soc.
ISBN: 9780821897454
Category : Mathematics
Languages : en
Pages : 132

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Book Description
The 1970s saw the appearance and development in categoricity theory of a tendency to focus on the study and description of uncountably categorical theories in various special classes defined by natural algebraic or syntactic conditions. There have thus been studies of uncountably categorical theories of groups and rings, theories of a one-place function, universal theories of semigroups, quasivarieties categorical in infinite powers, and Horn theories. In Uncountably Categorical Theories , this research area is referred to as the special classification theory of categoricity. Zilber's goal is to develop a structural theory of categoricity, using methods and results of the special classification theory, and to construct on this basis a foundation for a general classification theory of categoricity, that is, a theory aimed at describing large classes of uncountably categorical structures not restricted by any syntactic or algebraic conditions.

Models, Logics, and Higher-dimensional Categories

Models, Logics, and Higher-dimensional Categories PDF Author: Bradd T. Hart
Publisher: American Mathematical Soc.
ISBN: 0821883828
Category : Mathematics
Languages : en
Pages : 440

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Book Description
Proceedings of a conference held at Centre de recherches mathematiques of the Universite de Montreal, June 18-20, 2009.

Picturing Quantum Processes

Picturing Quantum Processes PDF Author: Bob Coecke
Publisher: Cambridge University Press
ISBN: 1108107710
Category : Science
Languages : en
Pages : 847

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Book Description
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.

New Spaces in Physics

New Spaces in Physics PDF Author: Mathieu Anel
Publisher: Cambridge University Press
ISBN: 110849062X
Category : Mathematics
Languages : en
Pages : 437

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Book Description
In this graduate-level book, leading researchers explore various new notions of 'space' in mathematical physics.