  Recently, the Hilbert-Schmidt speed, as a special class of quantum
statistical speed, has been reported to improve the interferometric phase in
single-parameter quantum estimation. Here, we test this concept in the
multiparameter scenario where two laser phases are estimated in a theoretical
model consisting of a three-level atom interacting with two classical
monochromatic fields. When the atom is initially prepared in the lower bare
state taking into account the detuning parameters, we extract an exact
analytical solution of the atomic density matrix in the case of two-photon
resonant transition. Further, we compare the performance of laser phase
parameters estimation in individual and simultaneous metrological strategies,
and we explore the role of quantum coherence in improving the efficiency of
unknown multi-phase shift estimation protocols. The obtained results show that
the Hilbert-Schmidt speed detects the lower bound on the statistical estimation
error as well as the optimal estimation regions, where its maximal corresponds
to the maximal quantum Fisher information, the performance of simultaneous
multiparameter estimation with individual estimation inevitably depends on the
detuning parameters of the three-level atom, and not only the quantum
entanglement, but also the quantum coherence is a crucial resource to improve
the accuracy of a metrological protocol.
