  New techniques based on weak measurements have recently been introduced to
the field of quantum state reconstruction. Some of them allow the direct
measurement of each matrix element of an unknown density operator and need only
$O(d)$ different operations, compared to $d^2$ linearly independent projectors
in the case of standard quantum state tomography, for the reconstruction of an
arbitrary mixed state. However, due to the weakness of these couplings, these
protocols are approximated and prone to large statistical errors. We propose a
method which is similar to the weak measurement protocols but works regardless
of the coupling strength: our protocol is not approximated and thus improves
the accuracy and precision of the results with respect to weak measurement
schemes. We experimentally apply it to the polarization state of single photons
and compare the results to those of preexisting methods for different values of
the coupling strength. Our results show that our method outperforms previous
proposals in terms of accuracy and statistical errors.
