Non-Demolition Measurement and Control in Quantum Dynamical Systems

  • Viacheslav Belavkin
Part of the International Centre for Mechanical Sciences book series (CISM, volume 294)


A multi-stage version of the theory of quantum-mechanical measurements and quantum-statistical decisions applied to the non-demolition control problem for quantum objects is developed. It is shown that in Gaussian case of quantum one-dimensional linear Markovian dynamical system with a quantum linear transmission line optimal quantum multistage decision rule consists of classical linear optimal control strategy and quantum optimal filtering procedure, the latter contains the optimal quantum coherent measurement on the output of the line and the recursive processing by Kalman-Busy filter. All the results are illustrated by an example of the optimal problem solution for a quantum one-dimensional linear oscillator on the input of a quantum wave transmission line.


Quantum Oscillator Optimal Quantum Optimal Observation Coherent Measurement Quantum Stochastic Differential Equation 
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  1. 1.
    Belavkin, V.P.: To the theory of control in quantum observable systems, Automic and Remove Control, 2(1983).Google Scholar
  2. 2.
    Davies, E.B. and J.T. Lewis: An operational approach to quantum probability, Commun. Math. Phys., 65(1970), 239–260.ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Accardi, L.: On the non-commutative Markov property, Functional Anal. Appl., 9(1975), 1–8.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Belavkin, V.P.: An operational theory of quantum stochastic processes, Proc. of VII-th Conference on Coding and Inform.Trans.Theory, Moscow-Vilnus, 1978.Google Scholar
  5. 5.
    Lindblad, G.: Non-Markovian quantum stochastic processes, Commun.Math.Phys., 65(1979), 281–294.ADSCrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Accardi, L. and Frigerio A. and J.T. Lewis: Quantum stochastic processes, Publ. RIMS Kyoto Univ., 18(1982), 97–133.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Barchielli, L. Lanz and G.M. Prosperi, Nuovo Cimento, B 72(1982), 79.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Belavkin, V.P.: Reconstruction theorem for quantum stochastic process, Theor. and Math.Phys., 62(1985), 409–431.MathSciNetGoogle Scholar
  9. 9.
    Butkovski, A.G. and U.J. Samoilenko: Control of quantum mechanical processes, Nauka, Moscow 1984.Google Scholar
  10. 10.
    Belavkin, V.P.: Optimal quantum filtering of Markov signals, Problems of Control and Inform. Theory, 7(1978), 345–360.MathSciNetGoogle Scholar
  11. 11.
    Belavkin, V.P.: Optimal filtration of Markovian signals in white quantum noise, Radio Eng. Electron. Phys., 18(1980), 1445–1453.MathSciNetGoogle Scholar
  12. 12.
    Accardi, L. and K.R. Parthasaraty: Quantum stochastic calculung. Springer LNM (to appear).Google Scholar
  13. 13.
    Belavkin, V.P.: Optimal measurement and control in quantum dynamical systems, Preprint N411, UMK, Torun, 1979.Google Scholar
  14. 14.
    Belavkin, V.P.: Optimization of quantum observation and control, Lecture Notes in Control and Inform. Sciences, IFIP, Optimization Techniques, Warsaw, 1979, Parti, Springer-Verlag.Google Scholar
  15. 15.
    Holevo, A.S.: Investigations on general statistical decision theory, Proc. of MIAN, CXXIV, Nauka, Moscow, 1976.Google Scholar
  16. 16.
    Aström, K.J.: Introduction to stochastic control theory, Academic Press, New York, 1970.MATHGoogle Scholar
  17. 17.
    Klauder, J.R. and E.C.D. Sudarshan:: Fundamentals of quantum optics. W.A.Benjamin, inc. New York, Amsterdam 1968.Google Scholar
  18. 18.
    Haus, H.A.: Steady-state quantum analysis of linear systems. Proc. IEEE, vol. 58, pp. 110–129, 1970.MathSciNetGoogle Scholar
  19. 19.
    Lax, M.: Quantum noise IV. Quantum theory of noise sources. Phys.Pev., vol. 145, pp. 1599–1611, 1965.Google Scholar
  20. 20.
    Belavkin, V.P.: Optimal linear randomized filtration of quantum boson signals. Problems of control and inform. theory, vol.3 (1), pp. 47–62, 1974.MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1987

Authors and Affiliations

  • Viacheslav Belavkin
    • 1
  1. 1.Miem, MoscowU.S.S.R.

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