Fundamental Tests of Quantum Geometric Bounds in Ionic and Covalent Insulators using Inelastic X-Ray Scattering

Quantum geometry underlies many fundamental properties of materials, but it has remained largely inaccessible to direct experiment. Here we demonstrate that inelastic x-ray scattering (IXS) provides a direct, quantitative probe of quantum geometry and quantum information in solids. Studying two prototype insulators, covalently bonded diamond and ionically bonded LiF, we measure the density response and experimentally determine the quantum Fisher information, the associated Bures metric, and the electron localization length. These measurements enable a quantitative comparison of quantum geometry for two distinct bonding environments. We find that the dimensionless quantum weight, $aK(q)$, which quantifies the longitudinal localization of quantum information, is constrained by fundamental electrostatic bounds in both materials. Crucially, the quantum weight of diamond exceeds that of LiF, indicating that covalent bonds exhibit a higher degree of delocalization and higher density of quantum information than the ionic bonds. Our results establish a direct experimental relationship between quantum information, electron localization, and chemical bonding, and identify IXS as a powerful tool for measuring quantum geometry in materials.