In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms. Whereas the theory described by the normal category of Hilbert spaces, Hilb, is ordinary quantum mechanics, the corresponding theory on finite dimensional Hilbert spaces is called fdQM.[1]
Properties
editThis 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.[2][3] 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
edit- ^ Kapustin, Anton (2013). "Is there life beyond Quantum Mechanics?". arXiv:1303.6917 [quant-ph].
- ^ Selinger, P. (2012) [2008]. "Finite dimensional Hilbert spaces are complete for dagger compact closed categories". Logical Methods in Computer Science. 8 (3). arXiv:1207.6972. CiteSeerX 10.1.1.749.4436. doi:10.2168/LMCS-8(3:6)2012.
- ^ Hasegawa, M.; Hofmann, M.; Plotkin, G. (2008). "Finite Dimensional Vector Spaces Are Complete for Traced Symmetric Monoidal Categories". In Avron, A.; Dershowitz, N.; Rabinovich, A. (eds.). Pillars of Computer Science. Vol. 4800. Lecture Notes in Computer Science: Springer. pp. 367–385. CiteSeerX 10.1.1.443.3495. doi:10.1007/978-3-540-78127-1_20. ISBN 978-3-540-78127-1.
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