  Multi-photon system has been studied by many groups, however the biggest
challenge faced is the number of copies of an unknown state are limited and far
from detecting quantum entanglement. The difficulty to prepare copies of the
state is even more serious for the quantum state tomography. One possible way
to solve this problem is to use adaptive quantum state tomography, which means
to get a preliminary density matrix in the first step and revise it in the
second step. In order to improve the performance of adaptive quantum state
tomography, we develop a new distribution scheme of samples and extend it to
three steps, that is to correct it once again based on the density matrix
obtained in the traditional adaptive quantum state tomography. Our numerical
results show that the mean square error of the reconstructed density matrix by
our new method is improved to the level from $10^{-4}$ to $10^{-9}$ for several
tested states. In addition, PhaseLift is also applied to reduce the required
storage space of measurement operator.
