Dagger compact category
From Wikipedia, the free encyclopedia
Dagger compact categories were introduced by John Baez and James Dolan[1] to describe topological quantum field theories. The formalism presented is sufficiently rich to capture the structure needed by some quantum information protocols namely: teleportation, logic gate teleportation and entanglement swapping[2]. Note that this particular description of those protocols take place in a dagger compact category with biproducts.
Contents |
[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 symmetric monoidal closed categories with duals for objects and morphisms[1]. Coecke and Abramsky shortened that to strongly compact closed categories. Later[3] they have been called dagger compact closed categories. Finally, the name dagger compact has been used in the most recent papers on this subject[4][5].
[edit] References
- ^ a b John C. Baez and James Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory, J.Math.Phys. 36 (1995) 6073-6105
- ^ 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).
- ^ 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).
- ^ 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).
- ^ 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.
