Dynamical realizability of kinematical bounds on the optimization of observables for quantum systems

Girardeau et al. (1998) derived kinematical bounds on the optimization of observables for mixed-state quantum systems and showed that they are dynamically realizable if the system is completely controllable. In this paper the problem of finding dynamically realizable bounds for systems that are not completely controllable is addressed. We derive such bounds for systems whose dynamics can be decomposed into subspace dynamics. We also study systems that are not decomposable yet fail to be completely controllable. For these systems, the question of dynamical realizability of the kinematical bounds depends on the accessibility of the target states for which the expectation value of the observable assumes its kinematical maximum.