Studia Logica

, Volume 103, Issue 4, pp 781–805 | Cite as

Duality for the Logic of Quantum Actions

  • Jort M. Bergfeld
  • Kohei Kishida
  • Joshua Sack
  • Shengyang Zhong


In this paper we show a duality between two approaches to represent quantum structures abstractly and to model the logic and dynamics therein. One approach puts forward a “quantum dynamic frame” (Baltag et al. in Int J Theor Phys, 44(12):2267–2282, 2005), a labelled transition system whose transition relations are intended to represent projections and unitaries on a (generalized) Hilbert space. The other approach considers a “Piron lattice” (Piron in Foundations of Quantum Physics, 1976), which characterizes the algebra of closed linear subspaces of a (generalized) Hilbert space. We define categories of these two sorts of structures and show a duality between them. This result establishes, on one direction of the duality, that quantum dynamic frames represent quantum structures correctly; on the other direction, it gives rise to a representation of dynamics on a Piron lattice.


Quantum logic Piron lattice Modal logic Labelled transition system Duality Orthomodular lattice 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Jort M. Bergfeld
    • 1
  • Kohei Kishida
    • 2
  • Joshua Sack
    • 1
  • Shengyang Zhong
    • 1
  1. 1.ILLCUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Department of Computer ScienceUniversity of OxfordOxfordUK

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