×

zbMATH — the first resource for mathematics

Conditional expectations, conditional distributions, and a posteriori ensembles in generalized probability theory. (English) Zbl 0645.60007
Summary: A general probabilistic framework containing the essential mathematical structure of any statistical physical theory is reviewed and enlarged to enable the generalization of some concepts of classical probability theory. In particular, generalized conditional probabilities of effects and conditional distributions of observables are introduced and their interpretation is discussed in terms of successive measurements.
The existence of generalized conditional distributions is proved, and the relation to M. Ozawa’s [Publ. Res. Inst. Math. Sci. 21, 279-295 (1985; Zbl 0576.60005)] a posteriori states is investigated. Examples concerning classical as well as quantum probability are discussed.

MSC:
60A99 Foundations of probability theory
82B99 Equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bauer, H. (1972).Probability Theory and Elements of Measure Theory, Holt, Rinehart and Winston, New York. · Zbl 0243.60004
[2] Cycon, H., and Hellwig, K.-E. (1977). Conditional expectations in generalized probability theory,Journal of Mathematical Physics,18, 1154-1161. · Zbl 0364.60007
[3] Davies, E. B. (1976).Quantum Theory of Open Systems, Academic Press, London. · Zbl 0388.46044
[4] Davies, E. B., and Lewis, J. T. (1970). An operational approach to quantum probability,Communications in Mathematical Physics,17, 239-260. · Zbl 0194.58304
[5] Edwards, C. M. (1970). The operational approach to algebraic quantum theory I,Communications in Mathematical Physics,16, 207-230. · Zbl 0187.25601
[6] Edwards, C. M. (1971). Classes of operations in quantum theory,Communications in Mathematical Physics,20, 26-56. · Zbl 0203.57001
[7] Gudder, S. P. (1979a). Axiomatic operational quantum mechanics,Reports on Mathematical Physics,16, 147-166. · Zbl 0437.46058
[8] Gudder, S. P. (1979b).Stochastic Methods in Quantum Mechanics, North-Holland, Amsterdam. · Zbl 0439.46047
[9] Gudder, S. P., and Marchand, J.-P. (1977). Conditional expectations on von Neumann algebras: A new approach,Reports on Mathematical Physics,12, 317-329. · Zbl 0379.60099
[10] Haag, R., and Kastler, D. (1964). An algebraic approach to quantum field theory,Journal of Mathematical Physics,5, 848-861. · Zbl 0139.46003
[11] Hellwig, K.-E. (1981). Conditional expectations and duals of instruments, inGrundlagen der Exakten Naturwissenschaften, Vol.5, H. Neumann, ed., pp. 113-124, Bibliographisches Institut, Mannheim. · Zbl 0537.60033
[12] Hellwig, K.-E., and Stulpe, W. (1983). A formulation of quantum stochastic processes and some of its properties,Foundations of Physics,13, 673-699.
[13] Kraus, K. (1983).States, Effects, and Operations, Lecture Notes in Physics190, Springer-Verlag, Berlin.
[14] Ludwig, G. (1983).Foundations of Quantum Mechanics I. Springer-Verlag, New York. · Zbl 0509.46057
[15] Ludwig, G. (1985).An Axiomatic Basis for Quantum Mechanics, Vol. I,Derivation of Hubert Space Structure, Springer-Verlag, Berlin. · Zbl 0582.46065
[16] Nagel, R. J. (1974). Order unit and base norm spaces, inFoundations of Quantum Mechanics and Ordered Linear Spaces, A. Hartkaemper and H. Neumann, eds., pp.29, 23-29, Springer-Verlag, Berlin.
[17] Nakamura, M., and Umgaki, H. (1962). On von Neumann’s theory of measurement in quantum statistics,Mathematica Japonica,7, 151-157. · Zbl 0113.09803
[18] Ozawa, M. (1985a). Concepts of conditional expectations in quantum theory,Journal of Mathematical Physics,26, 1948-1955. · Zbl 0568.60004
[19] Ozawa, M. (1985b). Conditional probability anda posteriori states in quantum mechanics,Publications RIMS Kyoto University,21, 279-295. · Zbl 0576.60005
[20] Stulpe, W. (1986). Bedingte Erwartungen und stochastische Prozesse in der generalisierten Wahrscheinlichkeitstheorie-Beschreibung sukzessiver Messungen mit zufälligem Ausgang, Doctoral dissertation, Berlin.
[21] Stulpe, W. (1987). Conditional expectations and stochastic processes in quantum probability, inInformation, Complexity, and Control in Quantum Physics, A. Blaquiere, S. Diner, and G. Lochak, eds., pp. 223-234, CISM Courses and Lectures No.294, Springer-Verlag, Vienna.
[22] Umegaki, H. (1954). Conditional expectations in an operator algebra I,Tohoku Mathematics Journal,6, 177-181. · Zbl 0058.10503
[23] Umegaki, H. (1964). Conditional expectations in an operator algebra IV,Kodai Mathematics Seminar Report,14, 59-85. · Zbl 0199.19706
[24] Werner, R. (1983). Physical uniformities on the state space of nonrelativistic quantum mechanics,Foundations of Physics,13, 859-881.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.