Abstract
The present paper is devoted to modelling of a probability measure of logical connectives on a quantum logic (QL), via a G-map, which is a special map on it. We follow the work in which the probability of logical conjunction, disjunction and symmetric difference and their negations for non-compatible propositions are studied.
We study such a G-map on quantum logics, which is a probability measure of a projection and show, that unlike classical (Boolean) logic, probability measure of projections on a quantum logic are not necessarilly pure projections.
We compare properties of a G-map on QLs with properties of a probability measure related to logical connectives on a Boolean algebra.
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Notes
- 1.
It is easy to see that if \(a\leftrightarrow b\), then \(q_p(a,b)=m_p(a)+m_p(b)-m_p(a\wedge b)=m_p(a\vee b)\) which explains its name.
- 2.
If \(a\leftrightarrow b\), then \(d(a,b)= m_d(a\bigtriangleup b)=m_d(a\wedge b')+m_d(a'\wedge b)\), where \(m_d\) is a state induced by d.
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Acknowledgement
The author (O. Nánásiová) would like to thank for the support of the VEGA grant agency by means of grant VEGA 1/0710/15 and VEGA 1/0159/17 and the author (L. Valášková) would like to thank for the support of VEGA 1/0420/15.
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Nánásiová, O., Čerňanová, V., Valášková, Ľ. (2020). Probability Measures and Projections on Quantum Logics. In: Kulczycki, P., Kacprzyk, J., Kóczy, L., Mesiar, R., Wisniewski, R. (eds) Information Technology, Systems Research, and Computational Physics. ITSRCP 2018. Advances in Intelligent Systems and Computing, vol 945. Springer, Cham. https://doi.org/10.1007/978-3-030-18058-4_25
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