Skip to main content
SpringerLink
Log in

Stochastic differential equations anda posteriori states in quantum mechanics

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is a“trajectory”for one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements one can ask what is the state of the system given that a certain trajectory up to timet has been observed. The response to this question is the notion ofa posteriori states and a“filtering”equation governing the evolution of such states: this turns out to be a nonlinear stochastic differential equation for density matrices or for pure vectors. The driving noise appearing in such an equation is not an external one, but its probability law is determined by the system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Barchielli, A. (1983).Nuovo Cimento,74B, 113–138.

    Google Scholar 

  • Barchielli, A. (1985).Physical Review D,32, 347–367.

    Google Scholar 

  • Barchielli, A. (1986a).Physical Review A,34, 1642–1649.

    Google Scholar 

  • Barchielli, A. (1986b). InFundamental Aspects of Quantum Theory, V. Gorini and A. Frigerio, eds., Plenum Press, New York, pp. 65–73.

    Google Scholar 

  • Barchielli, A. (1987).Journal of Physics A: Mathematical and General,20, 6341–6355.

    Google Scholar 

  • Barchielli, A. (1988). InQuantum Probability and Applications III, L. Accardi and W. von Waldenfels, eds., Springer, Berlin, pp. 37–51.

    Google Scholar 

  • Barchielli, A. (1990).Quantum Optics,2, 423–441.

    Google Scholar 

  • Barchielli, A. (1991). InQuantum Aspects of Optical Communications, C. Bendjaballah, O. Hirota, and S. Reynaud, eds., Springer, Berlin, pp. 179–189.

    Google Scholar 

  • Barchielli, A. (1993). InClassical and Quantum Systems—Foundations and Symmetries, H. D. Doebner, W. Scherer, and F. Schroeck, Jr., eds., World Scientific, Singapore, p. 488.

    Google Scholar 

  • Barchielli, A., and Belavkin, V. P. (1991).Journal of Physics A: Mathematical and General,24, 1495–1514.

    Google Scholar 

  • Barchielli, A., and Holevo, A. S. (n.d.). Constructing quantum processes via classical stochastic calculus, preprint.

  • Barchielli, A., and Lupieri, G. (1985a).Journal of Mathematical Physics 26, 2222–2230.

    Google Scholar 

  • Barchielli, A., and Lupieri, G. (1985b). InQuantum Probability and Applications II, L. Accardi and W. von Waldenfels, eds., Springer, Berlin, pp. 57–66.

    Google Scholar 

  • Barchielli, A., Lanz L., and Prosperi, G. M. (1982).Nuovo Cimento,72B, 79–121.

    Google Scholar 

  • Barchielli, A., Lanz, L., and Prosperi, G. M. (1983).Foundations of Physics,13, 779–812.

    Google Scholar 

  • Barchielli, A., Lanz, L., and Prosperi, G. M. (1985). InChaotic Behavior in Quantum Systems, G. Casati, ed., Plenum Press, New York, pp. 321–344.

    Google Scholar 

  • Belavkin, V. P. (1988). InModelling and Control of Systems, A. Blaquière, ed., Springer, Berlin, pp. 245–265.

    Google Scholar 

  • Belavkin, V. P. (1989a).Physics Letters A,140, 355–358.

    Google Scholar 

  • Belavkin, V. P. (1989b).Journal of Physics A: Mathematical and General,22, L1109-L1114.

    Google Scholar 

  • Belavkin, V. P. (1990a).Journal of Mathematical Physics,31, 2930–2934.

    Google Scholar 

  • Belavkin, V. P. (1990b).Letters in Mathematical Physics,20, 85–89.

    Google Scholar 

  • Belavkin, V. P. (1990c). InProbability Theory and Mathematical Statistics, B. Grigelioniset al., eds., Mokslas, Vilnius, and VSP, Utrecht, Volume 1, pp. 91–109.

    Google Scholar 

  • Belavkin, V. P. (1992).Communications in Mathematical Physics,146, 611–635.

    Google Scholar 

  • Belavkin, V. P., and Staszewski, P. (1989).Physics Leiters A,140, 359–362.

    Google Scholar 

  • Belavkin, V. P., and Staszewski, P. (1991).Reports on Mathematical Physics,29, 213–225.

    Google Scholar 

  • Carmichael, H. (1993).An Open Systems Approach to Quantum Optics, Springer, Berlin.

    Google Scholar 

  • Chruściński, D., and Staszewski, P. (1992).Physica Scripta,45, 193–199.

    Google Scholar 

  • Davies, E. B. (1969).Communications in Mathematical Physics,15, 277–304.

    Google Scholar 

  • Davies, E. B. (1970).Communications in Mathematical Physics,19, 83–105.

    Google Scholar 

  • Davies, E. B. (1971).Communications in Mathematical Physics,22, 51–70.

    Google Scholar 

  • Davies, E. B. (1976).Quantum Theory of Open Systems, Academic Press, London.

    Google Scholar 

  • Davies, E. B., and Lewis, J. T. (1970).Communications in Mathematical Physics,17, 239–260.

    Google Scholar 

  • Diósi, L. (1988a).Physics Letters A,129, 419–423.

    Google Scholar 

  • Diósi, L. (1988b).Physics Letters A,132, 233–236.

    Google Scholar 

  • Diósi, L. (1989).Physical Review A,40, 1165–1174.

    Google Scholar 

  • Dum, R., Zoller, P., and Ritsch, H. (1992a).Physical Review A,45, 4879–4887.

    Google Scholar 

  • Dum, R., Parkins, A. S., Zoller, P., and Gardiner, C. W. (1992b).Physical Review A,46, 4382–4396.

    Google Scholar 

  • Gardiner, C. W., Parkings, A. S., and Zoller, P. (1992).Physical Review A,46, 4363–4381.

    Google Scholar 

  • Gatarek, D., and Gisin, N. (1991).Journal of Mathematical Physics,32, 2152–2157.

    Google Scholar 

  • Ghirardi, G. C., Pearle, P., and Rimini, A. (1989). InProceedings 3rd International Symposium on the Foundations of Quantum Mechanics, Tokyo, pp. 181–189.

  • Ghirardi, G. C., Pearle, P., and Rimini, A. (1990a).Physical Review A,42, 78–89.

    Google Scholar 

  • Ghirardi, G. C., Pearle, P., and Rimini, A. (1990b). InSixty-Two Years of Uncertainty, A. I. Miller, ed., Plenum Press, New York, pp. 167–191.

    Google Scholar 

  • Ghirardi, G. C., Grassi, R., and Pearle, P. (1991). InSymposium on the Foundations of Modem Physics 1990, P. Lahti and P. Mittelstaedt, eds., World Scientific, Singapore, pp. 109–123.

    Google Scholar 

  • Gisin, N. (1984).Physics Review Letters,52, 1657–1660.

    Google Scholar 

  • Gisin, N. (1986).Journal of Physics A: Mathematical and General,19, 205–210.

    Google Scholar 

  • Gisin, N. (1989).Helvetica Physica Acta,62, 363–371.

    Google Scholar 

  • Gisin, N. (1990).Helvetica Physica Acta,63, 929–939.

    Google Scholar 

  • Gisin, N., and Percival, I. C. (1992a).Physics Letters A,167, 315–318.

    Google Scholar 

  • Gisin, N., and Percival, I. C. (1992b).Journal of Physics A: Mathematical and General,25, 5677–5691.

    Google Scholar 

  • Holevo, A. S. (1982).Soviet Mathematics,26, 1–20 [Izvestiya Vysshikh Uchebnykh Zavedenii. Seriya Matematika,26, 3–19].

    Google Scholar 

  • Holevo, A. S. (1988). InQuantum Probability and Applications III, L. Accardi and W. von Waldenfels, eds., Springer, Berlin, pp. 128–148.

    Google Scholar 

  • Holevo, A. S. (1989). InQuantum Probability and Applications IV, L. Accardi and W. von Waldenfels, eds., Springer, Berlin, pp. 229–255.

    Google Scholar 

  • Holevo, A. S. (1991). InQuantum Aspects of Optical Communications, C. Bendjaballahet al., eds., Springer, Berlin, pp. 127–137.

    Google Scholar 

  • Hudson, R. L., and Parthasarathy, K. R. (1984).Communications in Mathematical Physics,93, 301–323.

    Google Scholar 

  • Lindblad, G. (1976).Communications in Mathematical Physics,48, 119–130.

    Google Scholar 

  • Ludwig, G. (1967).Communications in Mathematical Physics,4, 331–348.

    Google Scholar 

  • Ludwig, G. (1970).Deutung des Begriffs“physikalische”Theorie und axiomatische Grundlegung der Hilbertraumstruktur der Quantenmechanik durch Hauptsätze des Messes, Springer, Berlin.

    Google Scholar 

  • Lupieri, G. (1983).Journal of Mathematical Physics,24, 2329–2339.

    Google Scholar 

  • Mølmer, K., Castin, Y., and Dalibard, J. (1993).Journal of the Optical Society of America,B10, 524–538.

    Google Scholar 

  • Nicrosini, O., and Rimini, A. (1990).Foundations of Physics,20, 1317–1327.

    Google Scholar 

  • Ozawa, M. (1984).Journal of Mathematical Physics,25, 79–87.

    Google Scholar 

  • Ozawa, M. (1985).Publications RIMS, Kyoto University,21, 279–295.

    Google Scholar 

  • Pearle, P. (1986).Physical Review D,33, 2240

    Google Scholar 

  • Srinivas, M. D. and Davies, E. B. (1981).Optica Acta,28, 981–996.

    Google Scholar 

  • Srinivas, M. D., and Davies, E. B. (1982).Optica Acta,29, 235–238.

    Google Scholar 

  • Staszewski, P., and Staszewska, G. (1992).Open Systems Information Dynamics,1, 103–114.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica dell'Università di Milano, I-20133, Milan, Italy

    A. Barchielli

  2. Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Milan, Italy

    A. Barchielli

Authors
  1. A. Barchielli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barchielli, A. Stochastic differential equations anda posteriori states in quantum mechanics. Int J Theor Phys 32, 2221–2233 (1993). https://doi.org/10.1007/BF00672994

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00672994

Keywords

Search

Navigation