Singular quantum Fisher information matrix in critical metrology

Critical metrology relies on the extreme sensitivity of the system's eigenstates close to the critical point to Hamiltonian parameter perturbations. Typically, however, the critical point, at which the phase transition occurs, is a function of many if not all the parameters of the system. This suggests that the quantum Fisher information matrix might be singular at the critical point, which in the context of parameter estimation is typically representing a significant complication. On the example of a toy-model Landau-Zener Hamiltonian, the Ising Hamiltonian, and the thermodynamic limit of the Lipkin-Meshkov-Glick Hamiltonian, we show that the quantum Fisher information matrix in critical metrology is always singular regardless of the system size and the distance to the critical point. However, contrary to the regular approach to metrology, where the singularity is detrimental, we argue that critical metrology works precisely because the quantum Fisher information matrix is singular.