  We study theoretically the usefulness of spin-1 Bose condensates with
macroscopic magnetization in a homogeneous magnetic field for quantum
metrology. We demonstrate Heisenberg scaling of the quantum Fisher information
for states in thermal equilibrium. The scaling applies to both
antiferromagnetic and ferromagnetic interactions. The effect preserves as long
as fluctuations of magnetization are sufficiently small. Scaling of the quantum
Fisher information with the total particle number is derived within the
mean-field approach in the zero temperature limit and exactly in the high
magnetic field limit for any temperature. The precision gain is intuitively
explained owing to subtle features of the quasi-distribution function in phase
space.
