Abstract
A description is presented of the indefinite structure of quantum stochastic (QS) calculus in Fock space, as developed by Hudson and Parthasarathy, with the quantum stochastic integral defined as a continuous operator on the projective limit of Fock spaces. Differential conditions are found for QS calculus of input-output QS processes and nondemolition measurements, and it is proved that the nondemolition condition is necessary and sufficient for the existence of conditional expectations relative to the subalgebra of observables and any state vector. A stochastic calculus of posterior (conditional) expectations of quantum nondemolition processes is developed, and a general stochastic equation is derived for quantum nonlinear filtering, both in the Heisenberg picture (for posterior operators) and in the Schrödinger picture (for the posterior density matrix and wavefunction). It is shown that posterior dynamics, unlike prior dynamics, does not mix states if the nondemolition measurement is complete.
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Literature cited
V. P. Belavkin, “Operational approach in the theory of quantum random processes of measurement and control,” in: Proc. VIIIth All-Union Conf. Theory of Coding and Transmission of Information, 1978 [in Russian], Part 1, Moscow-Vil'nyus (1978), pp. 23–28.
V. P. Belavkin, “Quantum filtering of Markov signals on a background of white quantum noise,” Radiotekh. Elektron.,25, No. 7, 1445–1453 (1980).
V. P. Belavkin, “On the theory of control of quantum observable processes,” Avtom. Telemekh., No. 2, 50–63 (1983).
V. P. Belavkin, “Theory of reconstruction for a quantum random process,” Teor. Mat. Fiz.,62, No. 3, 409–431 (1985).
V. P. Belavkin, “Nonlinear filtering of quantum continuous signals,” in: Proc. IXth All-Union Conf. Theory of Coding and Transmission of Information, 1988 [in Russian], Part 2, Odessa (1988), pp. 342–345.
V. B. Braginskii, Yu. I. Vorontsov, and D. I. Khalili, “Quantum properties of ponderomotive measurement of electromagnetic energy,” Zh. Teor. Eksp. Fiz.,73, 1340–1343 (1977).
R. Sh. Liptser and A. N. Shiryaev, Statistics of Random Processes [in Russian], Nauka, Moscow (1974).
A. V. Skorokhod, “Operator stochastic differential equations and stochastic semigroups,” Usp. Mat. Nauk,37, No. 6, 157–183 (1982).
R. L. Stratonovich, Conditional Markov Processes and Their Applications to Optimal Control [in Russian], Moscow State University, Moscow (1966).
A. S. Kholevo, “On the principle of quantum nondemolition measurements,” Teor. Mat. Fiz.,65, No. 3, 415–422 (1985).
A. Barchielli, “Input and output channels in quantum systems and quantum stochastic differential equations,” in: Proc. Oberwolf 1987 “Quantum Probability and Applications. III,” L. Accardi and W. von Waldenfels (eds.), Springer, Berlin (1988), pp. 37–51.
V. P. Belavkin, “Optimal measurement and control in quantum dynamical systems,” Preprint No. 411, 1979, Inst. Phz. Uniwersytet Mikolaja Kopernica, Torun.
V. P. Belavkin, “Nondemolition measurement and control in quantum dynamical systems,” in: Proc. CISM, Udein 1985, “Information Complexity and Control in Quantum Physics,” A. Blaquiere, S. Diner, and G. Loshak (eds.), Springer, Wien-New York (1987), pp. 311–336.
V. P. Belavkin, “Nondemolition stochastic calculus, nonlinear filtering and optimal control in open quantum systems,” in: Stochastic Methods in Mathematics of Physics, Proc. XXIVth Karpacz Winter School, World Scientific, Singapore (1988).
V. P. Belavkin, “Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes,” Proc. INRIA, Sophia-Antipolis 1988, “Bellman Contrinuum,” Springer (1988).
E. B. Davies and J. T. Lewis, “An operational approach to quantum probability,” Commun. Math. Phys.,17, No. 3, 239–260 (1970).
C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum statistical differential equations and the master equation,” Phys. Rev.,A31, 3761–3774 (1985).
R. S. Hudson and K. R. Parthasarathy, “Quantum Itô's formula and stochastic evolution,” Commun. Math. Phys.,93, No. 3, 301–323 (1984).
M. Lax, “Quantum noise IV. Quantum theory of noise sources,” Phys. Rev.,145, 110–129 (1965).
M. Lindsay and H. Maassen, “An integral kernel approach to noise,” in: Proc. Oberwolf 1987 “Quantum Probability and Applications. III,” L. Accardi and W. von Waldenfels (eds.), Springer, Berlin (1988), pp. 192–208.
M. Takesaki, “Conditional expectations on operator algebras,” J. Funct. Anal.,9, No. 3, 306–326 (1972).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 29–67, 1990.
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Belavkin, V.P. Stochastic calculus of quantum input-output processes and quantum nondemolition filtering. J Math Sci 56, 2625–2647 (1991). https://doi.org/10.1007/BF01095974
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DOI: https://doi.org/10.1007/BF01095974