Quantum entanglement and quantum geometry measured with inelastic X-ray scattering

Using inelastic X-ray scattering (IXS), we experimentally investigate the quantum geometry and quantum information in the large-gap insulator, LiF. Using sum rules for the density-density response function measured in IXS, we compute the quantum Fisher information of the equilibrium density matrix of LiF associated with density perturbations. Next, by exploiting universal relations between the quantum Fisher information, the optical conductivity, and the quantum metric tensor, we extrapolate the diagonal $(h,k,l) = (1,0,0)$ component of the quantum metric of LiF, known as the quantum weight. We compare our results to recently-proposed bounds on the quantum weight and find that the quantum weight in LiF comes close to saturating a theoretical upper bound, showing that quantum-mechanical delocalization plays an important role even in ionic insulators. Our work serves as a proof-of-principle that IXS techniques can be used to quantify state-of-the-art quantum geometric quantities of materials, and establishes the quantum Fisher information as an experimentally-accessible generalization of quantum geometry to real materials.