Jiaxin Liu, Zuoxian Wang, Danyue Ma

Engineered loss, gain, and non-reciprocity can sharply amplify a sensor's response, and exceptional points and the skin effect are widely proposed as routes to more precise quantum sensors. Some predict an exponentially large quantum Fisher information, the quantity that sets the best achievable precision, yet whether this amplified response is genuine measurement information has remained unsettled. We prove that it is not. For any open sensor whose parameter is imprinted coherently by a Hamiltonian, with loss, gain, non-reciprocity, and readout held fixed, the quantum Fisher information is capped by a single quantity, the spread between the largest and smallest energy the generator can impart, independent of the non-Hermitian dynamics. Amplification enlarges the response but not the information. The reported exponential gains arise only when that spread is unbounded, and any finite spread, such as a photon-number cutoff, restores the cap. The same limit governs dissipative probe preparation and spectral singularities. For photonic sensors the cap takes an operator form in which the information of any probe, classical or entangled, is set by the photon's dwell time at the perturbation, and it contains the recently derived scattering limits as special cases. Non-Hermiticity thus shapes a sensor's dynamics but does not source its precision, and the generator's spread and the attainable dwell time benchmark future non-Hermitian sensing proposals.