Hidden Fisher Information in Many-Body Quantum Sensors

arXiv:2610.33893v1 Announce Type: new Abstract: Quantum Fisher information (QFI) determines the ultimate precision of a quantum sensor after optimization over all allowed measurements, but it does not quantify how difficult the optimal readout is to implement. We introduce a moment Fisher hierarchy $F_M^{(k)}$, which measures the fraction of QFI accessible to collective spin moments up to degree $k$, together with a readout-complexity scale $k_{90}$ defined by the minimum degree required to recover $90\%$ of the QFI. Applying this framework to a nonlinear collective-spin sensor, we show that many-body dynamics can preserve large QFI while transferring the optimal readout into high-rank $SU(2)$ tensor sectors inaccessible to low-order collective observables.