Dagger compact category
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Dagger compact categories were introduced by B. Coecke and Samson Abramsky in [1] in order to recast the standard axiomatization of quantum mechanics in a category theoretic context.
The formalism presented in [1] is sufficiently rich to capture the structure needed by some quantum information protocols namely: teleportation, logic gate teleportation and entanglement swapping. Note that this particular description of those protocols take place in a dagger compact category with biproducts.
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[edit] Formal definition
In mathematics, a dagger compact category is a dagger symmetric monoidal category 
which is also compact closed and such that for all A in ,
commutes.
[edit] Examples
The following categories are dagger compact.
- The category FdHilb of finite dimensional Hilbert spaces and linear maps.
- The category Rel of Sets and relations.
- The category of finitely generated projective modules over a commutative ring.
[edit] Other appellations
Dagger compact categories were initially called strongly compact closed categories in [1]. In [4], they have been called dagger compact closed categories. Finally, the name dagger compact has been used in the most recent papers on this subject i.e. [2] and [3].
[edit] References
[1] S. Abramsky and B. 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).
[2] B. Coecke and E. O. Paquette, POVMs and Naimark's theorem without sums, to appear in: Proceedings of the 4th International Workshop on Quantum Programming Languages, Oxford (2005).
[3] B. Coecke and D. Pavlovic, Quantum measurements without sums, invited paper to appear in: The Mathematics of Quantum Computation and Technology; Chen, Kauffman and Lomonaco (eds.); Taylor and Francis.
[4] P. Selinger, Dagger compact closed categories and completely positive maps, Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30 - July 1 (2005).
