  We report an experimental realization of adaptive Bayesian quantum state
tomography for two-qubit states. Our implementation is based on the adaptive
experimental design strategy proposed in [F.Husz\'ar and N.M.T.Houlsby,
Phys.Rev.A 85, 052120 (2012)] and provides an optimal measurement approach in
terms of the information gain. We address the practical questions, which one
faces in any experimental application: the influence of technical noise, and
behavior of the tomographic algorithm for an easy to implement class of
factorized measurements. In an experiment with polarization states of entangled
photon pairs we observe a lower instrumental noise floor and superior
reconstruction accuracy for nearly-pure states of the adaptive protocol
compared to a non-adaptive. At the same time we show, that for the mixed states
the restriction to factorized measurements results in no advantage for adaptive
measurements, so general measurements have to be used.
