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.
