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.

1 Formal definition
2 Examples
3 Other appellations
4 References

[edit] Formal definition

In mathematics, a dagger compact category is a dagger symmetric monoidal category $\mathbb{C}$ which is also compact closed and such that for all $A$ in $\mathbb{C}$ ,

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).