On stochastic differential equations and semigroups of probability operators in quantum probability

Stochastic Processes and their Applications

Volumn 73, Issue 1, 1998, Pages 69-86

  Barchielli, A ^a   Paganoni, A M ^b   Zucca, F ^b  

^a POLITECNICO DI MILANO   (Italy)

^b UNIVERSITY OF MILAN   (Italy)

Author keywords

A posteriori states; Evolution systems; Instruments; Master equations; Semigroups of probability operators; Stochastic differential equations in Hilbert spaces

Indexed keywords

EID: 0032518007     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(97)00093-8     Document Type: Article

References (30)

1. Barchielli, A., Holevo, A,S., 1995. Constructing quantum measurement processes via classical stochastic calculus. Stochastic Process. Appl. 58, 293-317.
2. Barchielli, A., Holevo, A.S., Lupieri, G., 1993. An analogue of Hunt's representation theorem in quantum probability. J. Theoret. Probab. 6, 231-265.
3. Barchielli, A., Lanz, L., Prosperi, G.M., 1983. Statistic of continuous trajectories in quantum mechanics: operation valued stochastic processes. Found. Phys. 13, 779-812.
4. Barchielli, A., Lupieri, G., 1985. Quantum stochastic calculus, operation-valued stochastic processes, and continual measurements in quantum mechanics. J. Math. Phys. 26, 2222-2230.
5. Barchielli, A., Lupieri, G., 1991. A quantum analogue of Hunt's representation theorem for the generators of convolution semigroups on Lie groups. Probab. Theory Rel. Fields 88, 167-194.
6. Barchielli, A., Paganoni, A.M., 1996. A note on a formula of the Lévy-Khinchin type in quantum probability. Nagoya Math. J. 141, 29-43.
7. Belavkin, V.P., 1988. Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes. In: Blaquière, A. (Ed.), Modelling and Control of Systems, Lecture Notes in Control and Information Sciences, Vol. 121, Springer, Berlin, pp. 245-265.
8. Belavkin, V.P., 1989a. A new wave equation for a continuous nondemolition measurement. Phys. Lett. A 140, 355-358.
9. Belavkin, V.P., 1989b. A continuous counting observation and posterior quantum dynamics. J. Phys. A 22, L1109-L1114.
10. Belavkin, V.P., 1992. Quantum continual measurements and a posteriori collapse on CCR. Commun. Math. Phys. 146, 611-635.
11. Carmichael, H. (Ed.), 1996. Special Issue on Stochastic Quantum Optics, Quantum Semiclass. Opt. 8, 45-314.
12. Da Prato, G., Zabczyk, J., 1992. Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge.
13. Davies, E.B., 1976. Quantum Theory of Open Systems. Academic Press, New York.
14. Davies, E.B., Lewis, J.T., 1970. An operational approach to quantum probability. Commun. Math. Phys. 17, 239-260.
15. Diósi, L., 1988a. Continuous quantum measurement and Ito formalism, Phys. Lett. A 129, 419-423.
16. Diósi, L., 1988b. Localized solution of a simple nonlinear quantum Langevin equation. Phys. Lett. A 132, 233-236.
17. Ghirardi, G.C., Pearle, P., Rimini, A., 1990. Markov processes in Hilbert space and continuous sponataneous localization of systems of identical particles. Phys. Rev. A 42, 78-79.
18. Gisin, N., 1984. Quantum measurement and stochastic processes. Phys. Rev. Lett. 52, 1657-1660.
19. Gisin, N., Percival, I.C., 1992. The quantum-state diffusion model applied to open systems. J. Phys. A. 25, 5677-5691.
20. Holevo, A.S., 1996. On dissipative stochastic equations in a Hilbert space. Probab. Theory Rel. Fields 104, 483-500.
21. Ikeda, N., Watanabe, S., 1981. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
22. Lindblad, G., 1976. On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119-130.
23. Métivier, M., 1982. Semimartingales, a Course on Stochastic Processes. W. de Gruyter, Berlin.
24. Métivier, M., Pellaumail, J., 1980. Stochastic integration. Academic Press, New York.
25. Ozawa, M., 1984. Quantum measuring processes of continuous observables. J. Math. Phys. 25, 79-87.
26. Ozawa, M., 1985. Conditional probability and a posteriori states in quantum mechanics. Publ. R.I.M.S. Kyoto Univ. 21, 279-295.
27. Pazy, A., 1983. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin.
28. Reed, M., Simon, B., 1972. Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York.
29. Sakai, S., 1971. C*-algebras and W*-algebras. Springer, Berlin.
30. Wiseman, H.M., Milburn, G.J., 1993. Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere. Phys. Rev. A 47, 1652-1666.