Concept
Dagger compact category
Related concepts (6)
Dagger category
In , a branch of mathematics, a dagger category (also called involutive category or category with involution) is a equipped with a certain structure called dagger or involution. The name dagger category was coined by Peter Selinger. A dagger category is a category equipped with an involutive contravariant endofunctor which is the identity on . In detail, this means that: for all morphisms , there exist its adjoint for all morphisms , for all objects , for all and , Note that in the previous definition, the term "adjoint" is used in a way analogous to (and inspired by) the linear-algebraic sense, not in the category-theoretic sense.
Dagger symmetric monoidal category
In the mathematical field of , a dagger symmetric monoidal category is a that also possesses a . That is, this category comes equipped not only with a tensor product in the sense but also with a , which is used to describe unitary morphisms and self-adjoint morphisms in : abstract analogues of those found in FdHilb, the . This type of was introduced by Peter Selinger as an intermediate structure between and the that are used in categorical quantum mechanics, an area that now also considers dagger symmetric monoidal categories when dealing with infinite-dimensional quantum mechanical concepts.
Categorical quantum mechanics
Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably . The primitive objects of study are physical processes, and the different ways that these can be composed. It was pioneered in 2004 by Samson Abramsky and Bob Coecke. Categorical quantum mechanics is entry 18M40 in MSC2020. Mathematically, the basic setup is captured by a : composition of morphisms models sequential composition of processes, and the tensor product describes parallel composition of processes.
Compact closed category
In , a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the having finite-dimensional vector spaces as s and linear maps as s, with tensor product as the structure. Another example is , the category having sets as objects and relations as morphisms, with .
Symmetric monoidal category
In , a branch of mathematics, a symmetric monoidal category is a (i.e. a category in which a "tensor product" is defined) such that the tensor product is symmetric (i.e. is, in a certain strict sense, naturally isomorphic to for all objects and of the category). One of the prototypical examples of a symmetric monoidal category is the over some fixed field k, using the ordinary tensor product of vector spaces.
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 theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by Wootters and Zurek as well as Dieks the same year).