  When an informationally complete measurement is not available, the
reconstruction of the density operator that describes the state of a quantum
system can be accomplish, in a reliable way, by adopting the maximum entropy
principle (MaxEnt principle), as an additional criterion, to obtain the least
biased estimation. In this paper, we study the performance of the MaxEnt method
for quantum state estimation when there is prior information about symmetries
of the unknown state. We explicitly describe how to work with this method in
the most general case, and present an algorithm that allows to improve the
estimation of quantum states with arbitrary symmetries. Furthermore, we
implement this algorithm to carry out numerical simulations estimating the
density matrix of several three-qubit states of particular interest for quantum
information tasks. We observed that, for most states, our approach allows to
considerably reduce the number of independent measurements needed to obtain a
sufficiently high fidelity in the reconstruction of the density matrix.
Moreover, we analyze the performance of the method in realistic scenarios,
showing that it is robust even when considering the effect of finite
statistics, and under the presence of typical experimental noise.
