Abstract
This is the first of two papers concerned with the formulation of a continuous-time quantum-mechanical filter. Efforts focus on a quantum system with Hamiltonian of the formH 0+u(t)H 1, whereH 0 is the Hamiltonian of the undisturbed system,H 1 is a system observable which couples to an external classical field, andu(t) represents the time-varying signal impressed by this field. An important problem is to determine when and how the signalu(t) can be extracted from the time-development of the measured value of a suitable system observableC (invertibility problem). There exist certain quasiclassical observables such that the expected value and the measured value can be made to coincide. These are called quantum nondemolition observables. The invertibility problem is posed and solved for such observables. Since the physical quantum-mechanical system must be modelled as aninfinite-dimensional bilinear system, the domain issue for the operatorsH 0,H 1, andC becomes nontrivial. This technical matter is dealt with by invoking the concept of an analytic domain. An additional complication is that the output observableC is in general time-dependent.
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References
L. M. Silverman, Properties and application of inverse systems,IEEE Transactions on Automatic Control AC-13 (1968) 436–437.
J. L. Massey and M. K. Sain, Inverse of linear sequential circuits,IEEE Transactions on Computers 17 (1968) 330–337.
R. W. Brockett and M. D. Mesarovic, The reproducibility of multivariable control systems,Journal of Mathematical Analysis and Applications 11 (1965) 548–563.
B. D. O. Anderson and J. B. Moore, State estimation via the whitening filter, 1968, JACC, Ann Arbor, Michigan, pp. 123–129.
R. M. Hirschorn, Invertibility of nonlinear control systems,SIAM Journal of Control and Optimization 17 (1979) 289–297.
R. M. Hirschorn, Invertibility of multivariable nonlinear control systems,IEEE Transactions on Automatic Control AC-24 (1979) 855–865.
L. M. Silverman, Inversion of multivariable linear systems,IEEE Transactions on Automatic Control AC-19 (1969) 270–276.
D. Rebhuhn, Invertibility ofC ∞ multivariable input-output systems,IEEE Transactions on Automatic Control AC-25 (1980) 207–212.
H. Nijmeijer, Invertibility of affine nonlinear control systems: A geometric approach, preprint (to be published).
K. S. Thorne, C. M. Caves, V. D. Sandberg, M. Zimmermann and R. W. P. Drever, The quantum limit for gravitational-wave detectors and methods of circumventing it, inSources of Gravitational Radiation, edited by L. Smarr, p. 49. Cambridge University Press, Cambridge, England, 1979.
Vladimir B. Braginsky, Yuri I. Vorontsov and Kip S. Thorne, Quantum nondemolition measurements,Science, Vol. 209, 4456 (1980), pp. 547–557.
E. Nelson, Analytic vectors,Annals of Mathematics 70 (1959) 572–615.
A. Messiah,Quantum Mechanics, Vol. I. John Wiley and Sons, New York, 1962.
Claude Cohen-Tannoudji, Bernard Diu and Franck Laloë,Quantum Mechanics, Vol. 1. John Wiley and Sons, New York, 1977.
J. M. Jauch,Foundations of Quantum Mechanics. Addison-Wesley, New York, 1973.
T. J. Tarn, Garng Huang and John W. Clark, Modelling of quantum mechanical control systems,Journal of Mathematical Modelling 1 (1980) 109–121.
T. Kato,Perturbation Theory for Linear Operators. Springer Verlag, Berlin, 1976.
J. von Neumann,Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton, 1955.
A. O. Barut and R. Raczka,Theory of Group Representations and Applications. Polish Scientific Publishers, Warsaw, 1977.
H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems,Journal of Differential Equations 12 (1972) 95–116.
V. Jurdjevic and H. J. Sussmann, Control Systems on Lie groups,Journal of Differential Equations 12 (1972) 313–329.
Arthur J. Krener, A generalization of Chow's theorem and the bang-bang theorem to nonlinear control problems,SIAM Journal of Control 12 (1974) 43–51.
Ralph Abraham, Jerrold E. Marsden and Tudor Ratiu,Manifolds, Tensor Analysis, and Applications. Addison-Wesley, Reading, 1983.
Garng M. Huang, T. J. Tarn, and J. W. Clark, On the controllability of quantum-mechanical systems,Journal of Mathematical Physics 24 (1983) 2608–2618.
James Wei and Edward Norman, Lie algebraic solution of linear differential equations,Journal of Mathematical Physics 4 (1963) 575–581.
J. Wei and E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials,Proc. Am. Math. Soc. 15 (1964) 327–334.
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Research supported in part by the National Science Foundation under Grant Nos. ECS-8017184, INT-7902976 and DMR 8008229 and by the Department of Energy under Contract No. DE-AC01-79ET-29367.
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Ong, C.K., Huang, G.M., Tarn, T.J. et al. Invertibility of quantum-mechanical control systems. Math. Systems Theory 17, 335–350 (1984). https://doi.org/10.1007/BF01744448
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DOI: https://doi.org/10.1007/BF01744448