  The quantum Fisher information is of considerable interest not only for
quantum metrology but also because it is a useful entanglement measure for
finite temperature mixed states. In particular, it estimates the degree to
which multipartite entanglement is present. Recent results have related the
quantum Fisher information to experimentally measurable probes. While in
principle possible, a direct evaluation of the quantum Fisher information at
finite temperatures is technically challenging and here we show that a simple
estimate can be obtained for materials where the single mode approximation is
valid. We focus on the $S=1$ anti-ferromagnetic Heisenberg model with uniaxial
anisotropy. Quantum Monte Carlo thechniques are used to determine low
temperature correlations from which the quantum Fisher information can be
estimated within the single mode approximation. The quantum Fisher information
is compared to the quantum variance for the staggered magnetization operators
in the transverse direction and inequalities between the quantum Fisher
information, the quantum variance and the full variance are discussed. Both the
quantum and full variance as well as the quantum Fisher information are
examined at finite temperatures above the isotropic point and at the quantum
critical point for the Haldane-N\'{e}el transtion. A finite size scaling study
of the quantum Fisher information is performed at the quantum critical point
and used to confirm the Ising nature of the Haldane-N\'{e}el transition.

Strongly Correlated Electrons;Quantum Physics