QUANTUM FILTERING, DYNAMIC PROGRAMMING AND CONTROL

Quantum Filtering and Control (QFC)
as a dynamical theory of quantum feedback was initiated in my end of 70's papers and completed in the preprint [1]. This was my positive response to the general negative opinion that quantum systems have uncontrollable behavior in the process of measurement. As was showen in this and the following discrete [2] and continuous time [3] papers, the problem of quantum controllability is related to the problem of quantum observability which can be solved by means of quantum filtering. Thus the quantum dynamical filtering first was invented for the solution of the problems of quantum optimal control. The explicit solution [4] of quantum optimal linear filtering and control problem for quantum Gaussian Markov processes anticipated the general solution [5, 6] of quantum Markov filtering and control problems by quantum stochastic calculus technics. The derived in [5, 6] quantum nonlinear filtering equation for the posterior conditional expectations was represented also in the form of stochastic wave equations. The general solution of these filtering equations in the renormalized (in mean-square sense) linear form was constructed for bounded coefficients in [7] by quantum stochastic iterations. Quantum Filtering Theory (QFT) and the corresponding stochastic quantum equations have now wide applications for quantum continuos measurements, quantum dynamical reductions, quantum spontaneous localizations, quantum state diffusions, and quantum continuous trajectories. All these new quantum theories use particular types of stochastic Master equation which was initially derived from an extended quantum unitary evolution by quantum filtering method.
Quantum Programming and Computations (QPC)
is currently the hottest and most funding attractive topic of quantum cybernetics which in fact was launched under such name in Russia by R L Stratonovich in the beginning of 70's. It has become particularly popular after some speculative articles, the most notorious are the paper by R. Feynman (on quantum computers) and by D. Deutch (on quantum Turing machine). I followed another approach in quantum cybernetics, known as the alternative to Turing, based on the N. Wiener's ideas of finite automata. The computational process in this theory is represented as the dynamical programming process of feedback control in a dynamical system or net with finite state space, and the best computation is formulated in terms of optimal control. The probabilistic and quantum set up of this problem, involving the errors in state observation and noise perturbation, lead inevitably to the problems of error correction by classical or quantum noise filtering. Thus, if the automata is a quantum dynamical system, a net of quantum bits, say, the quantum computation process should be described by algorithms of quantum dynamical programming based on quantum dynamical filtering. There is no way in quantum world to observe and predict states of quantum automata, with the results of quantum computations, without errors and perturbations: quantum systems are probabilistic by their nature. This is why I believe that the quantum theory of dynamical programming, filtering and control [8--10] is the only theory giving the right approach to the quantum informational technologies of the new Century.
My 10 Relevant Publications:
  1. V. P. Belavkin: Optimal Measurement and Control in Quantum Dynamical Systems. Preprint Instytut Fizyki 411 3--38. Copernicus University, Torun', 1979. quant-ph/0208108, PDF.
  2. V. P. Belavkin: Optimization of Quantum Observation and Control. In Proc of 9th {IFIP} Conf on Optimizat Techn. Notes in Control and Inform Sci 1, Springer-Verlag, Warszawa 1979.
  3. V. P. Belavkin: Theory of the Control of Observable Quantum Systems. Automatica and Remote Control 44 (2) 178--188 (1983). quant-ph/0408003, PDF.
  4. V. P. Belavkin: Non-Demolition Measurement and Control in Quantum Dynamical Systems. In Information Complexity and Control in Quantum Systems 311--329, Springer--Verlag, Wien 1987.
  5. V. P. Belavkin: Non-Demolition Measurements, Nonlinear Filtering and Dynamic Programming of Quantum Stochastic Processes. Lecture notes in Control and Inform Sciences 121 245--265, Springer--Verlag, Berlin 1989.
  6. V. P. Belavkin: Non-Demolition Stochastic Calculus in Fock Space and Nonlinear Filtering and Control in Quantum Systems. In Stochastic Methods in Mathematics and Physics 310--324, World Scientific, Singapore 1989.
  7. V. P. Belavkin: Stochastic Equations of Quantum Filtering. In Prob. Theory and Math. Stat., 1, 91-109 B Grigelionis et al. (Eds) 1990 VSP/Mokslas
  8. V. P. Belavkin: Continuous Non-Demolition Observation, Quantum Filtering and Optimal Estimation. In Quantum Aspects of Optical Communication Lecture notes in Physics 45 131-145, Springer, Berlin 1991. quant-ph/0509205, PDF.
  9. V. P. Belavkin: Dynamical Programming, Filtering and Control of Quantum Markov Processes. In Stochastic Analysis in Mathematical Physics Lecture notes in Physics 9--42, World Scientific, Singapore 1998.
  10. V. P. Belavkin: Measurement, Filtering and Control in Quantum Open Dynamical Systems. Rep. on Math. Phys. 43 (3) 405-425 (1999).