  The article establishes a framework for dynamic generation of informationally
complete POVMs in quantum state tomography. Assuming that the evolution of a
quantum system is given by a dynamical map in the Kraus representation, one can
switch to the Heisenberg picture and define the measurements in the time
domain. Consequently, starting with an incomplete set of positive operators,
one can obtain sufficient information for quantum state reconstruction by
multiple measurements. The framework has been demonstrated on qubits and
qutrits. For some types of dynamical maps, it suffices to initially have one
measurement operator. The results demonstrate that quantum state tomography is
feasible even with limited measurement potential.
