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Statistical maps. II: Operational random variables and the Bell phenomenon. (English) Zbl 1088.81022

This paper is a continuation of the author’s discussion of statistical maps [S. Bugajski, Math. Slovaca 51, No. 3, 321–342 (2001; Zbl 1088.81021)]. It is first proved that any family of operational random variables having independent outcomes can be represented by a single standard random variable. Nevertheless, it is then demonstrated that certain families of operational random variables exhibit the Bell phenomenon which manifests itself in quantum mechanics as the well known Bell inequalities and is impossible in the framework of traditional probability theory. This indicates that operational random variables provide a nontrivial extension of traditional random variables.

MSC:

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
60A99 Foundations of probability theory

Citations:

Zbl 1088.81021
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References:

[1] BELL J. S.: On the Einstein-Podolsky-Rosen paradox. Physics 1 (1964), 195-200.
[2] FINE A.: Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48 (1982), 291-295.
[3] FINE A.: Joint distributions, quantum correlations, and commuting observables. J. Math. Phys. 23 (1982), 1306-1310.
[4] BELTRAMETTI E. G.-BUGAJSKI S.: The Bell phenomenon in classical frameworks. J. Phys. A 29 (1996), 247-261. · Zbl 0914.46060 · doi:10.1088/0305-4470/29/2/005
[5] ACCARDI L.-CECCHINI, C: Conditional expectations on von Neumann algebras and a theorem of Takesaki. J. Funct. Anal. 45 (1982), 245-273. · Zbl 0483.46043 · doi:10.1016/0022-1236(82)90022-2
[6] ACCARDI L.-FRIGERIO A.-LEWIS J. T.: Quantum stochastic processes. Publ. Res. Inst. Math. Sci. 18 (1982), 97-133. · Zbl 0498.60099 · doi:10.2977/prims/1195184017
[7] STREATER R. F.: Classical and quantum probability. arXiv:math-ph/0002029 (27 Feb 2000). · Zbl 0981.81006 · doi:10.1063/1.533322
[8] BUSCH P.- GRABOWSKI M.-LAHTI P. J.: Operational Quantum Physics. Springer-Verlag, Berlin, 1995. · Zbl 0863.60106 · doi:10.1007/978-3-540-49239-9
[9] BUGAJSKI S.: Fundamentals of fuzzy probability theory. Internat. J. Theoret. Phys. 35 (1996), 2229-2244. · Zbl 0872.60003 · doi:10.1007/BF02302443
[10] BUGAJSKI S.-HELLWIG K.-E.-STULPE W.: On fuzzy random variables and statistical maps. Rep. Math. Phys. 41 (1998), 1-11. · Zbl 1026.60501 · doi:10.1016/S0034-4877(98)80180-8
[11] BUGAJSKI S.: Fuzzy stochastic processes. Open Syst. Inf. Dyn. 5 (1998), 169-185. · Zbl 0908.60044 · doi:10.1023/A:1009673617619
[12] GUDDER S.: Fuzzy probability theory. Demonstratio Math. 31 (1998), 235-254. · Zbl 0984.60001
[13] BUGAJSKI S.: Statistical maps I. Basic properties. Math. Slovaca 51 (2001), 321 342. · Zbl 1088.81021
[14] BUGAJSKI S.: Classical and quantal in one or How to describe mesoscopic systems. Molecular Phys. Rep. 11 (1995), 161-171.
[15] PURI M. L.-RALESCU D. A.: Fuzzy random variables. J. Math. Anal. Appl. 114 (1986), 409-422. · Zbl 0605.60038 · doi:10.1098/rspa.1986.0091
[16] RIEČAN B.-NEUBRUNN T.: Integral, Measure, and Ordering. Math. Appl. 411, Kluwer, Dordrecht, 1997. · Zbl 0916.28001
[17] BAUER H.: Probability Theory and Elements of Measure Theory. Academic Press, London, 1981. · Zbl 0466.60001
[18] HOLEVO A. S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam, 1982. · Zbl 0497.46053
[19] BUSCH P.-LAHTI P. J.-MITTELSTAEDT P.: The Quantum Theory of Measurement. (2nd, Springer-Verlag, Berlin, 1996. · Zbl 0868.46051 · doi:10.1007/978-3-540-37205-9
[20] ARAKI H.: A remark on Machida-Namiki theory of measurement. Progr. Theoret. Phys. 64 (1980), 719-730. · Zbl 1097.81503 · doi:10.1143/PTP.64.719
[21] VON NEUMANN J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton, N.J., 1955. · Zbl 0064.21503
[22] MISRA B.: On a new definition of quantal states. Physical Reality and Mathematical Description (C P. Enz, J. Mehra, D. Reidel Publishing Company, Dordrecht-Holland, 1974, pp. 455-476.
[23] BELTRAMETTI E. G.-BUGAJSKI S.: Quantum observables in classical frameworks. Internat. J. Theoret. Phys. 34 (1995), 1221-1229. · Zbl 0850.81019 · doi:10.1007/BF00676232
[24] BELTRAMETTI E. G.-BUGAJSKI S.: A classical extension of quantum mechanics. J. Phys. A 28 (1995), 3329-3343. · Zbl 0859.46049 · doi:10.1088/0305-4470/28/12/007
[25] BOHM D.: Quantum Theory. Prentice-Hall, Inc, Englewood Cliffs, NJ., 1951. · Zbl 0048.21802
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