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Concrete Quantum Logics and Δ -Logics, States and Δ -States

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Abstract

By a concrete quantum logic (in short, by a logic) we mean the orthomodular poset that is set-representable. If \(L=({\Omega },\mathcal {L})\) is a logic and \(\mathcal {L}\) is closed under the formation of symmetric difference, Δ , we call L a Δ -logic. In the first part we situate the known results on logics and states to the context of Δ -logics and Δ -states (the Δ -states are the states that are subadditive with respect to the symmetric difference). Moreover, we observe that the rather prominent logic \(\mathcal {E}^{\text {even}}_{\Omega }\) of all even-coeven subsets of the countable set Ω possesses only Δ -states. Then we show when a state on the logics given by the divisibility relation allows for an extension as a state. In the next paragraph we consider the so called density logic and its Δ -closure. We find that the Δ -closure coincides with the power set. Then we investigate other properties of the density logic and its factor.

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Acknowledgements

The authors were supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS15/193/OHK3/3T/13.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27, Prague, Czech Republic

    Michal Hroch & Pavel Pták

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  1. Michal Hroch
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  2. Pavel Pták
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Correspondence to Michal Hroch.

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Hroch, M., Pták, P. Concrete Quantum Logics and Δ -Logics, States and Δ -States. Int J Theor Phys 56, 3852–3859 (2017). https://doi.org/10.1007/s10773-017-3359-x

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  • DOI: https://doi.org/10.1007/s10773-017-3359-x

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