mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

FdHilb has all finite-dimensional

Hilbert spaces In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

for

objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

and the

linear transformations In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

between them as

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

Properties

This category * is monoidal, * possesses finite

biproduct In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for ...

s, and * is dagger compact. According to a theorem of Selinger, the category of finite-dimensional Hilbert spaces is complete in the

dagger compact category In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from thei ...

. Many ideas from Hilbert spaces, such as the

no-cloning theorem In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theore ...

, hold in general for dagger compact categories. See that article for additional details.

References

Monoidal categories Dagger categories Hilbert spaces {{categorytheory-stub