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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.
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.
New scientific paradigms typically consist of an expansion of the
conceptual language with which we describe the world. Over the past
decade, theoretical physics and quantum information theory have
turned to category theory to model and reason about quantum
protocols. This new use of categorical and algebraic tools allows a
more conceptual and insightful expression of elementary events such
as measurements, teleportation and entanglement operations, that
were obscured in previous formalisms. Recent work in natural
language semantics has begun to use these categorical methods to
relate grammatical analysis and semantic representations in a
unified framework for analysing language meaning, and learning
meaning from a corpus. A growing body of literature on the use of
categorical methods in quantum information theory and computational
linguistics shows both the need and opportunity for new research on
the relation between these categorical methods and the abstract
notion of information flow. This book supplies an overview of how
categorical methods are used to model information flow in both
physics and linguistics. It serves as an introduction to this
interdisciplinary research, and provides a basis for future
research and collaboration between the different communities
interested in applying category theoretic methods to their domain's
open problems.
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