  Recent experiments have demonstrated the generation of entanglement by
quasi-adiabatically driving through quantum phase transitions of a
ferromagnetic spin-1 Bose-Einstein condensate in the presence of a tunable
quadratic Zeeman shift. We analyze, in terms of the Fisher information, the
interferometric value of the entanglement accessible by this approach. In
addition to the Twin-Fock phase studied experimentally, we unveil a second
regime, in the broken axisymmetry phase, which provides Heisenberg scaling of
the quantum Fisher information and can be reached on shorter time scales. We
identify optimal unitary transformations and an experimentally feasible optimal
measurement prescription that maximize the interferometric sensitivity. We
further ascertain that the Fisher information is robust with respect to
non-adiabaticity and measurement noise. Finally, we show that the
quasi-adiabatic entanglement preparation schemes admit higher sensitivities
than dynamical methods based on fast quenches.
