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 group from their category of finite-dimensional continuous unitary representations^[1]. They also appeared independently in the work of Baez and 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.

Selinger showed that dagger compact categories admit a Joyal-Street style diagrammatic language^[3] and proved that dagger compact categories are complete with respect to finite dimensional Hilbert spaces^[4] i.e. an equational statement in the language of dagger compact categories holds if and only if it can be derived in the concrete category of finite dimensional Hilbert spaces and linear maps.

Connection to quantum information processing

Dagger compact categories were applied to capture some quantum information protocols namely: teleportation, logic gate teleportation and entanglement swapping^[5][6]; the resulting line of research is now known as categorical quantum mechanics ^[7].

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.

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.

References

1. ^ S. Doplicher and J. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989) 157-218.
2. ^ J. C. Baez and J. Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory, J.Math.Phys. 36 (1995) 6073-6105
3. ^ 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).
4. ^ P. Selinger, Finite dimensional Hilbert spaces are complete for dagger compact closed categories, Proceedings of the 5th International Workshop on Quantum Programming Languages, Reykjavik (2008).
5. ^ 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).
6. ^ 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.
7. ^ 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).