Categorical Equivalence Between Orthomodular Dynamic Algebras and Complete Orthomodular Lattices
- 120 Downloads
This paper provides a categorical equivalence between two types of quantum structures. One is a complete orthomodular lattice, which is used for reasoning about testable properties of a quantum system. The other is an orthomodular dynamic algebra, which is a quantale used for reasoning about quantum actions. The result extends to more restrictive lattices than orthomodular lattices, and includes Hilbert lattices of closed subspaces of a Hilbert space. These other lattice structures have connections to a wide range of different quantum structures; hence our equivalence establishes a categorical connection between quantales and a great variety of quantum structures.
KeywordsOrthomodular dynamic algebra Complete orthomodular lattice Quantale
We would like to thank Professor Roberto Giuntini for his suggestion that we consider orthomodular lattices rather than just Hilbert lattices in the equivalence. Kishida’s research has been supported by the grants FA9550-12-1-0136 of the U.S. AFOSR and EP/N018745/1 of EPSRC. Rafiee Rad’s research is funded by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 283963. Zhong’s research is supported by NSSFC Grant 14ZDB015.
- 3.Bergfeld, J.M., Kishida, K., Sack, J., Zhong, S.: Duality for the logic of quantum actions. Studia Logica. 1–25 (2014). doi: 10.1007/s11225-014-9592-x
- 5.Rosenthal, K.: Quantales and Their Applications. Longman Higher Education (1990)Google Scholar
- 7.Stubbe, I., van Steirteghem, B.: Propositional systems, Hilbert lattices and generalized Hilbert spaces. In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures, pp 477–523. Elsevier Science B.V., Amsterdam (2007), 10.1016/B978-044452870-4/50033-9