Conditional probability and a posteriori states in quantum mechanics. (English) Zbl 0576.60005

Ozawa, Masanao

Publ. Res. Inst. Math. Sci. 21, 279-295 (1985).

Summary: In order to develop a statistical theory of quantum measurements including continuous observables, a concept of a posteriori states is introduced, which generalizes the notion of regular conditional probability distributions in classical probability theory. Its statistical interpretation in measuring processes is discussed and its existence is proved. As an application, we also give a complete proof of the E. B. Davies and J. T. Lewis [Commun. Math. Phys. 17, 239-260 (1970; Zbl 0194.583)] conjecture that there are no (weakly) repeatable instruments for non-discrete observables in the standard formulation of quantum mechanics, using the notion of a posteriori states.

MSC:

60A99	Foundations of probability theory
81P20	Stochastic mechanics (including stochastic electrodynamics)

Keywords:

statistical theory of quantum measurements; quantum mechanics

Citations:

Zbl 0194.583

References:

[1]	Araki, H., Yanase, M. M., Measurement of quantum mechanical operators. Phys. Rev. 120 (1960), 622-626. · Zbl 0095.42502
[2]	Arveson, W. B., Analyticity in operator algebras. Amer. J. Math. 89 (1967), 578-642. · Zbl 0183.42501
[3]	Berberian, S. K., Notes on spectral theory. Princeton, Van Nostrand, 1966. · Zbl 0138.39104
[4]	Breiman, L., Probability. London, Addison-Wesley, 1968. · Zbl 0174.48801
[5]	Cycon, H., Hellwig, K. -E., Conditional expectations in generalized probability theory. /. Math, Phys. 18 (1977), 1154-1161. · Zbl 0364.60007
[6]	Davies, E, B., Quantum theory of open systems. London, Academic Press, 1976. · Zbl 0388.46044
[7]	Davies, E. B., Lewis, J. T., An operational approach to quantum probability. Commun. Math. Phys. 17 (1970), 239-260. · Zbl 0194.58304
[8]	Holevo, A. S., Statistical decision theory for quantum systems. /. Multivar. Anal. 3 (1973), 337-394. · Zbl 0275.62004
[9]	Holevo, A. S., Probabilistic and statistical aspects of quantum theory. Amsterdam, North-Holland, 1982. · Zbl 0497.46053
[10]	Kraus, K., General state changes in quantum theory. Ann. Phys. 64 (1971), 311-335. · Zbl 1229.81137
[11]	Nakamura, M., Umegaki, H., On von Neumann’s theory of measurements in quantum statistics. Math. Japonica 1 (1962), 151-157. · Zbl 0113.09803
[12]	von Neumann, J., Mathematical foundations oj quantum mechanics. Princeton, Princeton Univ. Press, 1955. · Zbl 0064.21503
[13]	Ozawa, M., Quantum measuring processes of continuous observables. J. Math. Phys. 25 (1984), 79-87.
[14]	Schatten, R., Norm ideals of completely continuous operators. Berlin, Springer, 1960. · Zbl 0090.09402
[15]	Schwartz, L., Lectures on disintegration of measures. Bombay, Tata Institute of Fundamental Research, 1975.
[16]	Takesaki, M., Theory of operator algebras L New York, Springer, 1979. · Zbl 0436.46043
[17]	Tomiyama, J., On the projection of norm one in W^-algebra. Proc. Japan Acad. 33 (1957), 608-612. · Zbl 0081.11201
[18]	Tulcea, I. T., Tulcea, 1. C, Topics in the theory of lifting. New York, Springer, 1969. · Zbl 0179.46303
[19]	Umegaki, H., Conditional expectation in an operator algebra I, II, III, IV, Tohoku Math. J. 6 (1954), 177-181, 8 (1956), 86-100, Kodai Math. Sem. Rep. 11 (1959), 51-74, 14 (1962), 59-85. · Zbl 0072.12501
[20]	Wigner, E. P., Die Messung Quantenmechanischer Operatoren. Z. Phys. 133 (1952), 101-108. · Zbl 0048.44102
[21]	Davies, E. B., On the Borel structure of C*-algebras, Commun. Math. Phys. 8 (1959), 147-168, with Appendix by R. V. Kadison.