mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

FdHilb has all finite-dimensional

Hilbert spaces In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

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 ai ...

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 pr ...

between them as

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s. Whereas the theory described by the normal category of Hilbert spaces, Hilb, is ordinary quantum mechanics, the corresponding theory on finite dimensional Hilbert spaces is called fdQM.

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 fo ...

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 computer, quantum computing among ...

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

References

Categories in category theory Monoidal categories Dagger categories Hilbert spaces {{categorytheory-stub