Summary
In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of instruments on groups and the associated semigroups of probability operators. In this paper the case is considered of a finite-dimensional Hilbert space (n-level quantum system) and of instruments defined on a finite-dimensional Lie group. Then, the generator of a continuous semigroup of (quantum) probability operators is characterized. In this way a quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained.
References
Barchielli, A.: Convolution semigroups in quantum probability. Semsterbericht Funktionalanalysis Tübingen, Sommersemester 1987, Bd. 12, pp. 29–42
Barchielli, A.: Probability operators and convolution semigroups of instruments in quantum probability. Probab. Th. Rel. Fields82, 1–8 (1989)
Barchielli, A., Lupieri, G.: Quantum stochastic calculus, operation valued stochastic processes and continual measurements in quantum mechanics. J. Math. Phys.26, 2222–2230 (1985)
Barchielli, A., Lupieri, G.: Dilations of operation valued stochastic processes. In: Accardi, L., Waldenfels, W. von (eds.) Quantum probability and applications II. (Lect. Notes Math., vol. 1136, pp. 57–66) Berlin Heidelberg New York: Springer 1985
Barchielli, A., Lupieri, G.: Convolution semigroups in quantum probability and quantum stochastic calculus. In: Accardi, L., Waldenfels, W. von (eds.) Quantum probability and applications IV. (Lect. Notes Math., vol. 1396, pp. 107–127) Berlin Heidelberg New York: Springer 1989
Bourbaki, N.: XXIX, Eléments de mathématique, Livre VI, Intégration. Paris: Harmann 1963
Courrège, Ph.: Générateur infinitésimal d'un semi-groupe de convolution sur ℝn et formule de Lévy-Khinchine. Bull. Sci. Math. II. Sér.88, 3–30 (1964)
Davies, E.B.: Quantum theory of open systems. London: Academic Press 1976
Davies, E.B.: One-parameter semigroups. London: Academic Press 1980
Denisov, L.V., Holevo, A.S.: Conditionally positive definite functions on the coefficient algebra of a nonabelian group. Preprint, Steklov Mathematical Institute, Moscow (1989)
Dugundji, J. Topology. Boston: Allyn and Bacon 1970
Dynkin, E.B.: Markov processes. Vol. I. Berlin Göttingen Heidelberg: Springer 1965
Evans, D.E., Lewis, J.T.: Dilations of irreversible evolutions in algebraic quantum theory. Commun. Dublin Inst. Adv. Studies, Series A, N. 24 (1977)
Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups ofN-level systems. J. Math. Phys.17, 821–825 (1976)
Hewitt, E., Ross, K.A.: Abstract harmonic analysis. Vol. I. Berlin Göttingen Heidelberg: Springer 1963
Heyer, H.: Probability measures on locally compact groups. Berlin Heidelberg New York: Springer 1977
Holevo, A.S.: Infinitely divisble measurements in quantum probability. Teor. Veroyatn. Primen.31, 560–564 (1986) [Theory Probab. Appl.31, 493–497 (1987)]
Holevo, A.S.: A representation of Lévy-Khinchine type in quantum probability theory. Teor. Veroyatn. Primen.32, 142–146 (1987) [Theory Probab. Appl.32, 131–136 (1988)]
Holevo, A.S.: Conditionally positive definite functions in quantum probability. Proceedings of the International Congress of Mathematicians, Berkeley, California, USA, 1986, pp. 1011–1020
Holevo, A.S.: A noncommutative generalization of conditionally positive definite functions. In: Accardi, L., Waldenfels, W. von (eds.) Quantum probability and applications III. (Lect. Notes Math., vol. 1303, pp. 128–148) Berlin Heidelberg New York: Springer 1988
Holevo, A.S.: Limits theorems for repeated measurements and continuous measurement processes. In: Accardi, L., Waldenfels, W. von (eds.) Quantum probability and applications IV. (Lect. Notes Math., vol. 1396, pp. 229–255) Berlin Heidelberg New York: Springer 1989
Hunt, G.A.: Semi-groups of measures on Lie groups. Trans. Am. Math. Soc.81, 264–293 (1956)
Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys.48, 119–130 (1976)
Parthasarathy, K.R.: One parameter semigroups of completely positive maps on groups arising from quantum stochastic differential equations. Boll. Unione Mat. Ital., VII. Ser., A5, 391–397 (1986)
Ramaswami, S.: Semigroups of measures on Lie groups. J. Indian Math. Soc.38, 175–189 (1974)
Sakai, S.:C *-algebras andW *-algebras. Berlin Heidelberg New York: Springer 1971
Waldenfels, W. von: Fast positive Operatoren. Z. Wahrscheinlichkeitstheor. Verw. Geb.4, 159–174 (1965)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Barchielli, A., Lupieri, G. A quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups. Probab. Th. Rel. Fields 88, 167–194 (1991). https://doi.org/10.1007/BF01212558
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01212558
Keywords
- Hilbert Space
- Convolution
- Quantum Mechanic
- Probability Measure
- Probability Theory