  We demonstrate a novel experimental technique for quantum-state tomography of
the collective density matrix. It is based on measurements of the polarization
of light, traversing the atomic vapor. To assess the technique's robustness
against errors, experimental investigations are supported with numerical
simulations. This not only allows to determine the fidelity of the
reconstruction, but also to analyze the quality of the reconstruction for
specific experimental parameters light tuning and number of measurements). By
utilizing the so-called conditional number, we demonstrate that the
reconstruction can be optimized for a specific tuning of the system parameters,
and further improvement is possible by selective repetition of the
measurements. Our results underscore the potential high-fidelity quantum-state
reconstruction while optimizing measurement resources.
