Quantum Fisher information matrix in Heisenberg XY model

The quantum Fisher information matrix provides us with a tool to determine the precision, in any multiparametric estimation protocol, through quantum Cram\'er-Rao bound. In this work, we study simultaneous and individual estimation strategies using the density matrix vectorization method. Two special Heisenberg $XY$ models are considered. The first one concerns the anisotropic $XY$ model in which the temperature $T$ and the anisotropic parameter $\gamma$ are estimated. The second situation concerns the isotropic $XY$ model submitted to an external magnetic field $B$ in which the temperature and the magnetic field are estimated. Our results show that the simultaneous strategy of multiple parameters is always advantageous and can provide a better precision than the individual strategy in the multiparameter estimation procedures.