Dynamical quantum Fisher information in the Ising model

In terms of quantum Fisher information, we discuss the dynamics of quantity χ2 of the Ising model. The inequality χ2<1 detects the class of entangled states which are useful for sub-shot-noise sensitivity. All the exact dynamics of the expectations of collective spin operators are derived. The minimum χ2 in the plane perpendicular to the mean spin direction is analytically given. We find that χ2 does not depend on the system size N when N≥4. Except for the periodic points, the evolution states are always entangled and useful for the sub-shot-noise phase estimation.