A geometric perspective: experimental evaluation of the quantum Cramer-Rao bound

Quantum parameter estimation plays a central role in quantum metrology. Its ultimate precision limit is known as the quantum Cramer-Rao bound. In multi-parameter estimation, the quantum Cramer-Rao bound is usually not saturated due to incompatibility of observables in quantum physics. In this work, we explore the connection between multi-parameter estimation and quantum geometry, and provide experimental evaluation of the Cramer-Rao bound through geometric measurements. A concrete example of estimating two (three) parameters in a spin-1/2 (spin-1) system is analyzed in detail, where fundamental uncertainty principles prevent saturating the Cramer-Rao bound. We measure the quantum fisher information and Berry curvature, and for the first time, experimentally extract the attainable Cramer-Rao bound for three-parameter estimations.