No abstract

We consider a new class of dynamical systems whose state is represented by a Hermitian matrix motivated by treating quantum control problems. We develop theory and techniques of differential geometric analysis for dynamical systems in that class, where a Lie product of matrix functions is introduced and plays an important role. We provide a simple and coordinate-free calculation method for the Lie product of matrix functions which enables efficient differential geometric analysis. The result of this paper will be used in a follow-up paper on analysis of local state transition of controlled quantum systems under continuous quantum measurement with imperfect detector efficiency.

:In this paper, we analyze local state transition of controlled quantum systems under continuous quantum measurement, which is described by a matrix-valued nonlinear stochastic differential equation. To this end, we utilize the method of differential geometric analysis for systems with matrix-valued states developed in the first part of this series. The method provides us a direct and efficient way of analysis with a clear perspective. We study local state transition of the controlled quantum systems with imperfect detector efficiency which has not been studied enough in previous works. The controlled quantum system with perfect detector efficiency is also investigated as a special case. Sufficient conditions for the measurement operator and the control Hamiltonian are derived, under which the local state transition is quite limited. We also show that the conditions are satisfied in many typical situations. The results reveal fundamental nature of the controlled quantum systems under continuous quantum measurement.

No abstract