  In this paper, we explore the differences between classical logarithmic
fidelity and quantum fidelity. The classical logarithmic fidelity is found to
be always extensive while the quantum one manifests distinct size dependence in
different phases. Illustrated by the anisotropic Lipkin-Meshkov-Glick model, we
found numerically and analytically that the logarithmic fidelity scales like N
in the symmetry-broken phase and scales like N^0 in the polarized phase. The
singular behavior around the critical point is also investigated.
