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Categorical Equivalence Between Orthomodular Dynamic Algebras and Complete Orthomodular Lattices

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Abstract

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.

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Notes

  1. The existence of a join for every (potentially infinite) set guarantees the existence of a meetas well; the meet (greatest lower bound) of A is the join of the set of lower bounds of A.

  2. The original function \(\bigsqcup \)is then the join operation on the lattice structure,from which the order relation can be recovered by\(x\sqsubseteq y\)iff x ⊔ y = y.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    Kohei Kishida

  2. Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Soroush Rafiee Rad

  3. Department of Mathematics and Statistics, California State University Long Beach, Long Beach, CA, 90840, USA

    Joshua Sack

  4. Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China

    Shengyang Zhong

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  1. Kohei Kishida
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  2. Soroush Rafiee Rad
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  3. Joshua Sack
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  4. Shengyang Zhong
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Correspondence to Soroush Rafiee Rad.

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Kishida, K., Rafiee Rad, S., Sack, J. et al. Categorical Equivalence Between Orthomodular Dynamic Algebras and Complete Orthomodular Lattices. Int J Theor Phys 56, 4060–4072 (2017). https://doi.org/10.1007/s10773-017-3433-4

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  • DOI: https://doi.org/10.1007/s10773-017-3433-4

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