Probability Measures and Projections on Quantum Logics
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.
KeywordsLogical connectives Orthomodular lattice Quantum logic Probability measure State
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.
- 1.Nánásiová, O., Čerňanová, V., Valášková, Ľ.: Probability measures and projections on quantum logics. In: Kulczycki, P., Kowalski, P.A., Łukasik, S. (eds.) Contemporary Computational Science, p. 78. AGH-UST Press, Cracow (2018)Google Scholar
- 11.Khrennikov, A.: Violation of Bell’s inequality and non-Kolmogorovness. In: Accardi, L., et al. (eds.) Foundations of Probability and Physics-5. American Institute of Physics, Mellville (2009)Google Scholar
- 16.Nánásiová, O., Pykacz, J.: Modelling of uncertainty and bi-variable maps. J. Electr. Eng. 67(3), 169–176 (2016)Google Scholar
- 20.Pavičić, M.: Classical logic and quantum logic with multiple and common lattice models. Hindawi Publishing Corporation Advances in Mathematical Physics volume 2016, Article ID 6830685, 12 pages (2016)Google Scholar
- 23.Pitovsky, I.: Quantum Probability-Quantum Logic. Springer, Berlin (1989)Google Scholar