Robust quantum metrology using disordered probes

Disorder is ubiquitous in quantum devices including quantum probes designed and fabricated for quantum parameter estimation and sensing. We investigate the robustness of a quantum probe against the presence of glassy disorder. We define a disorder marker quantifying the effect of the disorder by expanding the quantum Fisher information in terms of different orders of the standardized central moments of the disorder-distributions. We classify the quantum probes in terms of the possible values of the disorder marker, and analytically show, for a disorder-sensitive probe with identical and weak disorder on all or a subset of the parameters of the probe-Hamiltonian, that the absolute value of the disorder marker exhibits a quadratic dependence on the disorder strength. We derive a robustness scale intrinsic to the probe that competes with the disorder, and provide a prescription for estimating the maximum disorder strength that the probe can withstand from the disorder-free probe-Hamiltonian for a given initial state of the probe, which can be computed without the disorder averaging. We demonstrate our results in the case of a single-qubit probe under disordered magnetic field, and a multi-qubit probe described by a disordered one-dimensional Kitaev model with nearest-neighbor interactions.