Dagger compact category

In mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Doplicher and Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations (that is, Tannakian categories). They also appeared in the work of Baez and Dolan as an instance of semistrict k-tuply monoidal n-categories, which describe general topological quantum field theories, for n = 1 and k = 3. They are a fundamental structure in Abramsky and 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 dagger symmetric monoidal category ${\displaystyle \mathbf {C} }$ which is also compact closed, 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 ${\displaystyle A}$ in ${\displaystyle \mathbf {C} }$, the following diagram commutes:

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