Sensitivity bounds for interferometry with Ising Hamiltonians

Sensitivity bounds for a generic interferometric phase estimation problem are the shot noise and the Heisenberg limits. The shot noise is the highest sensitivity that can be reached with separable states, while the Heisenberg limit is the ultimate bound in sensitivity which can be saturated with entangled states. The scaling of these bounds with the number of particles N entering the interferometer depends on the specific Hamiltonian governing the same interferometer. In typical cases, the Hamiltonian is linear, and the shot-noise and Heisenberg limits scale with N-1/2 and with N-1, respectively. With interferometers described by generic, nonlinear Hamiltonian, the scalings with the number of particles can be rather different. Here we study the shot-noise and Heisenberg limits of Ising-like Hamiltonians in the presence of longitudinal and transverse fields, both in the nearest-neighbor and in the fully connected spin interaction cases. We provide the explicit forms of the states saturating the shot-noise and Heisenberg limits. These results can be relevant not only for precision measurement purposes but also to characterize quantum phase transitions and, more generally, the witnessing of multipartite entanglement.