Quantum Inf Process (2013) 12:1539–1626
Quantum quasi-Markov processes in eventum
mechanics dynamics, observation, ﬁltering and control
Viacheslav P. Belavkin
Received: 23 November 2011 / Accepted: 23 July 2012 / Published online: 7 September 2012
© Springer Science+Business Media, LLC 2012
Abstract 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-indepen-
dent 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 measure-
ments to describe the stochastic evolution of the open quantum system under the
continuous indirect observation and working in parallel with classical indeterminis-
tic 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 explic-
itly solvable quantum linear-quadratic-Gaussian problem which emphasizes many
similarities with the corresponding classical control problem.
Keywords Quantum probability · Quantum stochastics · Quantum Trajectories ·
Conditionally-Markov dynamics · Quantum ﬁltering · Quantum feedback control
The author acknowledges the support under the programme ATESIT (contract no IST-2000-29681) from
the EC and also QBIC programme of Tokyo Science University where it was completed.
V. P. Belavkin (
School of Mathematical Sciences, University of Nottingham, Nottingham, UK