Quantum quasi-Markov processes in eventum mechanics dynamics, observation, filtering and control

• Authors: Viacheslav P. Belavkin
• Venue: Quantum Information Processing
• Year: 2013

Quantum mechanical systems exhibit an inherently probabilistic behavior upon measurement which excludes in principle the singular case of direct observability. The theory of quantum stochastic time continuous measurements and quantum filtering was earlier developed by the author on the basis of non-Markov conditionally-independent increment models for quantum noise and quantum nondemolition observability. Here this theory is generalized to the case of demolition indirect measurements of quantum unstable systems satisfying the microcausality principle. The exposition of the theory is given in the most general algebraic setting unifying quantum and classical theories as particular cases. The reduced quantum feedback-controlled dynamics is described equivalently by linear quasi-Markov and nonlinear conditionally-Markov stochastic master equations. Using this scheme for diffusive and counting measurements to describe the stochastic evolution of the open quantum system under the continuous indirect observation and working in parallel with classical indeterministic control theory, we derive the Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these measurements. The resulting Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian problem which emphasizes many similarities with the corresponding classical control problem.