Phase-Space Geometry and Optimal State Preparation in Quantum Metrology with Collective Spins

We revisit well-known protocols in quantum metrology using collective spins and propose a unifying picture for optimal state preparation based on a semiclassical description in phase space. We show how this framework allows for quantitative predictions of the timescales required to prepare various metrologically useful states, and that these predictions remain accurate even for moderate system sizes, surprisingly far from the classical limit. Furthermore, this framework allows us to build a geometric picture that relates optimal (exponentially fast) entangled probe preparation to the existence of separatrices connecting saddle points in phase space. We illustrate our results with the paradigmatic examples of the two-axis counter-twisting and twisting-and-turning Hamiltonians, where we provide analytical expressions for all the relevant optimal time scales. Finally, we propose a generalization of these models to include $p$-body collective interaction (or $p$-order twisting), beyond the usual case of $p=2$. Using our geometric framework, we prove a no-go theorem for the local optimality of these models for $p>2$.