International Journal of Theoretical Physics

, Volume 56, Issue 12, pp 4060–4072 | Cite as

Categorical Equivalence Between Orthomodular Dynamic Algebras and Complete Orthomodular Lattices

  • Kohei Kishida
  • Soroush Rafiee RadEmail author
  • Joshua Sack
  • Shengyang Zhong


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.


Orthomodular 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.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK
  2. 2.Institute for Logic, Language and ComputationUniversity of AmsterdamAmsterdamThe Netherlands
  3. 3.Department of Mathematics and StatisticsCalifornia State University Long BeachLong BeachUSA
  4. 4.Institute of Logic and CognitionSun Yat-sen UniversityGuangzhouChina

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