Indirect Measurement of Quantum Fisher Information

Quantum Fisher information sets the ultimate precision of quantum parameter estimation, yet measuring it generally requires full state tomography or an explicit reconstruction of the symmetric logarithmic derivative. We show that, in the short-time limit, it is fixed by a single variance, that of the operator generating the parameter shift, evaluated in the initial state. For two pure states evolving under the same Hamiltonian, the ratio of their quantum Fisher information equals the ratio of these variances, so the value for an unknown state follows from a characterized reference. When every initial state is a cavity state tensored with the atomic ground state, this ratio reduces to that of the short-time changes in the atomic energy, and projective measurements of the Pauli spin operator on the atom alone suffice. We apply the relation to the extended multiphoton Rabi and Jaynes–Cummings models. The numerics confirm the predicted agreement, quantify the finite-time deviations that bound the useful measurement window, and show how the reference state, through its mean photon number and statistical family, governs how long this agreement persists. The scheme needs no state reconstruction and no measurement optimization, and we quantify both the finite-time bias and the statistical resources required.