Category of finite-dimensional Hilbert spaces

In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms.

Properties

This category

is monoidal,
possesses finite biproducts, and
is dagger compact.

According to a theorem of Selinger, the category of finite-dimensional Hilbert spaces is complete in the dagger compact category.[1][2] Many ideas from Hilbert spaces, such as the no-cloning theorem, hold in general for dagger compact categories. See that article for additional details.

References

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).
M. Hasegawa, M. Hofmann and G. Plotkin, "Finite dimensional vector spaces are complete for traced symmetric monoidal categories", LNCS 4800, (2008), pp. 367–385.

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[1] 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).

[2] M. Hasegawa, M. Hofmann and G. Plotkin, "Finite dimensional vector spaces are complete for traced symmetric monoidal categories", LNCS 4800, (2008), pp. 367–385.