Density functional fidelity susceptibility and Kullback-Leibler entropy

It is shown that density functional fidelity susceptibility is approximately proportional to the relative or Kullback-Leibler entropy. A nearly linear relationship between the relative entropy and density functional fidelity is explored. The ratio of the fidelity and the distance of the relative entropy from 1 detects quantum phase transition. This is illustrated by numerical and analytical variational approximations for the ground state of the single-mode Dicke model.