Abstract
In this note we contribute to the recently developing study of “almost Boolean” quantum logics (i.e. to the study of orthomodular partially ordered sets that are naturally endowed with a symmetric difference). We call them enriched quantum logics (EQLs). We first consider set-representable EQLs. We disprove a natural conjecture on compatibility in EQLs. Then we discuss the possibility of extending states and prove an extension result for \(\mathbb {Z}_{2}\)-states on EQLs. In the second part we pass to general orthoposets with a symmetric difference (GEQLs). We show that a simplex can be a state space of a GEQL that has an arbitrarily high degree of noncompatibility. Finally, we find an appropriate definition of a “parametrization” as a mapping between GEQLs that preserves the set-representation.
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Acknowledgements
The second author acknowledges the support by the Austrian Science Fund (FWF):Project I 4579-N and the Czech Science Foundation (GACR):Project 20-09869L.
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Burešová, D., Pták, P. Quantum Logics that are Symmetric-difference-closed. Int J Theor Phys 60, 3919–3926 (2021). https://doi.org/10.1007/s10773-021-04950-6
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DOI: https://doi.org/10.1007/s10773-021-04950-6