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

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.