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Concept
Dagger compact category
Résumé
In , 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 their category of finite-dimensional continuous unitary representations (that is, ). They also appeared in the work of John Baez and James Dolan as an instance of semistrict k-tuply , which describe general topological quantum field theories, for n = 1 and k = 3. They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics.
Dagger compact categories can be used to express and verify some fundamental quantum information protocols, namely: teleportation, logic gate teleportation and entanglement swapping, and standard notions such as unitarity, inner-product, trace, Choi–Jamiolkowsky duality, complete positivity, Bell states and many other notions are captured by the language of dagger compact categories. All this follows from the completeness theorem, below. Categorical quantum mechanics takes dagger compact categories as a background structure relative to which other quantum mechanical notions like quantum observables and complementarity thereof can be abstractly defined. This forms the basis for a high-level approach to quantum information processing.
A dagger compact category is a which is also , together with a relation to tie together the dagger structure to the compact structure. Specifically, the dagger is used to connect the unit to the counit, so that, for all in , the following diagram commutes:
To summarize all of these points:
A category is if it has an internal hom functor; that is, if the hom-set of morphisms between two objects of the category is an object of the category itself (rather than of Set).
A category is if it is equipped with an associative bifunctor that is associative, natural and has left and right identities obeying certain coherence conditions.
Source officielle
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