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Probability measures and logical connectives on quantum logics

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EN
The present paper is devoted to modelling of a probabi‐ lity measure of logical connectives on a quantum logic via a G‐map, which is a special map on it. We follow the work in which the probability of logical conjunction (AND), dis‐ junction (OR), symmetric difference (XOR) and their nega‐ tions for non‐compatible propositions are studied. Now we study all remaining cases of G‐maps on quantum lo‐ gic, namely a probability measure of projections, of impli‐ cations, and of their negations. We show that unlike clas‐ sical (Boolean) logic, probability measures of projections on a quantum logic are not necessarilly pure projections. We indicate how it is possible to define a probability me‐ asure of implication using a G‐map in the quantum logic, and then we study some properties of this measure which are different from a measure of implication in a Boolean algebra. Finally, we compare the properties of a G‐map with the properties of a probability measure related to logical connectives on a Boolean algebra.
Twórcy
  • Inst. of Computer Science and Mathematics, Slovak University of Technology in Bratislava, Ilkovičova 3, 812 19 Bratislava, Slovakia, www: matika.elf.stuba.sk/KMAT/OlgaNanasiova
  • Department of Mathematics and Descriptive Geometry, Slovak University of Techno‑ logy , Radlinského 11, 810 05 Bratislava, Slovakia, www: math.sk
  • Department of Mathematics and Computer Science, Faculty of Education, Trna‑ va University, Priemyselná 4, 918 43 Trnava, Slovakia, www: pdf.truni.sk/katedry/kmi/pracovnici
Bibliografia
  • [1] A. M. Al‑Adilee and O. Nánásiová , “Copula and s‑map on a quantum logic”, Information Sciences, vol. 179, no. 24, 2009, 4199–4207, 10.1016/j.ins.2009.08.011.
  • [2] G. Birkhoff and J. Von Neumann, “The Logic of Quantum Mechanics”, Annals of Mathematics, vol. 37, no. 4, 1936, 823–843, 10.2307/1968621.
  • [3] M. Bohdalova and O. Nanasiova. “Note to Copula Functions”, 2006.
  • [4] L. J. Bunce, M. Navara, P. Ptá k, and J. D. M. Wright, “Quantum logics with Jauch‑Piron states”, The Quarterly Journal of Mathematics, vol. 36, no. 3, 1985, 261–271, 10.1093/qmath/36.3.261.
  • [5] E. Drobná , O. Nánásiová , and L. Valášková , “Quantum logics and bivariable functions”, Kybernetika, vol. 46, no. 6, 2010, 982–995.
  • [6] A. Dvurecenskij and S. Pulmannová , New Trends in Quantum Structures, Mathematics and Its Applications, Springer Netherlands, 2000, 10.1007/978‑94‑017‑2422‑7.
  • [7] A. Dvurečenskij and S. Pulmannová , “Connection between joint distribution and compatibility”, Reports on Mathematical Physics, vol.19, no. 3, 1984, 349–359, 10.1016/0034‑4877(84)90007‑7.
  • [8] L. Herman, E. L. Marsden, and R. Piziak, “Implicaction connectives in orthomodular lattices”, Notre Dame Journal of Formal Logic, vol. 16, no. 3, 1975,305–328, 10.1305/ndjl/1093891789.
  • [9] A. Y. Khrennikov, “EPR‑Bohm experiment and Bell’s inequality: Quantum physics meets probability theory”, Theoretical and Mathematical Physics, vol. 157, no. 1, 2008, 1448–1460, 10.1007/s11232‑008‑0119‑3.
  • [10] A. Khrennikov, “Violation of Bell’s Inequality and non‑Kolmogorovness”, Foundations of Probability and Physics‑5. AIP Conference Proceedings, vol. 1101, no. 1, 2009, 86–99, 10.1063/1.3109976.
  • [11] O. l. Nánásiová and L. Valášková , “Marginality and Triangle Inequality”,InternationalJournal of Theoretical Physics, vol. 49, no. 12, 2010, 3199–3208, 10.1007/s10773‑010‑0414‑2.
  • [12] O. Nánásiová , V. Cerňanová , and L. Valášková , “Probability Measures and Projections on Quantum Logics”. In: P. Kulczycki, J. Kacprzyk, L. T. Kóczy, R. Mesiar, and R. Wisniewski, eds., Information Technology, Systems Research, and Computational Physics, Cham, 2020, 321–330, 10.1007/978‑3‑030‑18058‑4_25.
  • [13] O. Nánásiová and L. Valášková , “Maps on a quantum logic”, Soft Computing, vol. 14, no. 10, 2010, 1047–1052, 10.1007/s00500‑009‑0483‑4.
  • [14] O. Nánásiová , “Map for Simultaneous Measurements for a Quantum Logic”, International Journal of Theoretical Physics, vol. 42, no. 9, 2003, 1889–1903, 10.1023/A:1027384132753.
  • [15] O. Nánásiová , “Principle Conditioning”, International Journal of Theoretical Physics, vol. 43, no. 7, 2004, 1757–1767,10.1023/B:IJTP.0000048818.23615.28.
  • [16] O. Nánásiová and M. Kalina, “Calculus for Non Compatible Observables, Construction Through Conditional States”, International Journal of Theoretical Physics, vol. 54, no. 2, 2015, 506–518, 10.1007/s10773‑014‑2243‑1.
  • [17] O. Nánásiová and A. Khrennikov, “Representation Theorem of Observables on a Quantum System”, International Journal of Theoretical Physics, vol. 45, no. 3, 2006, 469–482, 10.1007/s10773‑006‑9030‑6.
  • [18] O. Nánásiová and J. Pykacz, “Modelling of Uncetainty and Bi–Variable Maps”, Journal of Electrical Engineering, vol. 67, no. 3, 2016, 169–176, 10.1515/jee‑2016‑0024.
  • [19] M. Pavicic and N. D. Megill, “Is Quantum Logic a Logic?”, arXiv:0812.2698 [quant‑ph], 2008, arXiv: 0812.2698.
  • [20] M. Pavičić, “Exhaustive generation of orthomodular lattices with exactly one nonquantum state”, Reports on Mathematical Physics, vol. 64, no. 3, 2009, 417–428, 10.1016/S0034‑4877(10)00005‑4.
  • [21] M. Pavičić. “Classical Logic and Quantum Logic with Multiple and Common Lattice Models”, 2016.
  • [22] C. Piron and J. Jauch, “On the structure of quantal proposition systems”, Birkhäuser, 1969, 10.5169/seals‑114098.
  • [23] I. Pitowsky, Quantum Probability — Quantum Logic, Lecture Notes in Physics, Springer‑Verlag: Berlin Heidelberg, 1989, 10.1007/BFb0021186.
  • [24] P. Ptá k and S. Pulmannová , Orthomodular structures as quantum logics, number v. 44 in Fundamental theories of physics, Kluwer Academic Publishers: Dordrecht ; Boston, 1991.
  • [25] J. Pykacz and P. Frąckiewicz, “The Problem of Conjunction and Disjunction in Quantum Logics”, International Journal of Theoretical Physics, vol. 56, no. 12, 2017, 3963–3970, 10.1007/s10773‑017‑3402‑y.
  • [26] J. Pykacz, L. Valášková , and O. Nánásiová , “Bell Type Inequalities for Bivariate Maps on Orthomodular Lattices”, Foundations of Physics, vol. 45, no. 8, 2015, 900–913, 10.1007/s10701‑015‑9906‑5.
  • [27] G. Sergioli, G. M. Bosyk, E. Santucci, and R. Giuntini, “A Quantum‑inspired Version of the Classification Problem”, International Journal of Theoretical Physics, vol. 56, no. 12, 2017, 3880–3888, 10.1007/s10773‑017‑3371‑1.
  • [28] S. Sozzo, “Conjunction and negation of natural concepts: A quantum‑theoretic modeling”, Journal of Mathematical Psychology, vol. 66, 2015,83–102, 10.1016/j.jmp.2015.01.005.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-45aad990-1d45-4c89-8f5f-37d73215634b
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