Dagger compact category

[1] They also appeared in the work of John Baez and James Dolan as an instance of semistrict k-tuply monoidal n-categories, which describe general topological quantum field theories,[2] for n = 1 and k = 3.

They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics.

[3][4][5] 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.

This completeness result implies that various theorems from Hilbert spaces extend to this category.

The completeness theorem implies that basic notions from Hilbert spaces carry over to any dagger compact category.

Together, these obey five axioms:[11] Comultiplicativity: Coassociativity: Cocommutativity: Isometry: Frobenius law: To see that these relations define a basis of a vector space in the traditional sense, write the comultiplication and counit using bra–ket notation, and understanding that these are now linear operators acting on vectors

that can satisfy the above five axioms must be orthogonal to one-another; the counit then uniquely specifies the basis.

The suggestive names copying and deleting for the comultiplication and counit operators come from the idea that the no-cloning theorem and no-deleting theorem state that the only vectors that it is possible to copy or delete are orthogonal basis vectors.

Given the above definition of a basis, a number of results for Hilbert spaces can be stated for compact dagger categories.