Estimates of the Quantum Fisher Information in the $S=1$ Anti-Ferromagnetic Heisenberg Spin Chain with Uniaxial Anisotropy*

The quantum Fisher information has relevance to quantum metrology and as an entanglement measure. We focus on the $S=1$ anti-ferromagnetic Heisenberg model with uniaxial anisotropy.
Quantum Monte Carlo techniques 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 transition. 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.