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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).〔S. Doplicher and J. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989) 157-218.〕 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,〔J. C. Baez and J. Dolan, ''(Higher-dimensional Algebra and Topological Quantum Field Theory )'', J.Math.Phys. 36 (1995) 6073-6105〕 for n = 1 and k = 3. They are a fundamental structure in Abramsky and Coecke's categorical quantum mechanics.〔Samson Abramsky and Bob Coecke, ''( A categorical semantics of quantum protocols )'', Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004).〕〔S. Abramsky and B. Coecke, ''( Categorical quantum mechanics )". In: Handbook of Quantum Logic and Quantum Structures, K. Engesser, D. M. Gabbay and D. Lehmann (eds), pages 261–323. Elsevier (2009).〕〔Abramsky and Coecke used the term strongly compact closed categories, since a dagger compact category is a compact closed category augmented with a covariant involutive monoidal endofunctor.〕
==Overview==
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.
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