  Adaptive techniques have important potential for wide applications in
enhancing precision of quantum parameter estimation. We present a recursively
adaptive quantum state tomography (RAQST) protocol for finite dimensional
quantum systems and experimentally implement the adaptive tomography protocol
on two-qubit systems. In this RAQST protocol, an adaptive measurement strategy
and a recursive linear regression estimation algorithm are performed. Numerical
results show that our RAQST protocol can outperform the tomography protocols
using mutually unbiased bases (MUB) and the two-stage MUB adaptive strategy
even with the simplest product measurements. When nonlocal measurements are
available, our RAQST can beat the Gill-Massar bound for a wide range of quantum
states with a modest number of copies. We use only the simplest product
measurements to implement two-qubit tomography experiments. In the experiments,
we use error-compensation techniques to tackle systematic error due to
misalignments and imperfection of wave plates, and achieve about 100-fold
reduction of the systematic error. The experimental results demonstrate that
the improvement of RAQST over nonadaptive tomography is significant for states
with a high level of purity. Our results also show that this recursively
adaptive tomography method is particularly effective for the reconstruction of
maximally entangled states, which are important resources in quantum
information.
