Quantum estimation of the Schwarzschild spacetime parameters of the Earth

We propose a quantum experiment to measure with high precision the Schwarzschild spacetime parameters of the Earth. The scheme can also be applied to measure distances by taking into account the curvature of the Earth's spacetime. As a wave packet of (entangled) light is sent from the Earth to a satellite it is redshifted and deformed due to the curvature of spacetime. Measurements after the propagation enable the estimation of the spacetime parameters. We compare our results with the state of the art, which involves classical measurement methods, and discuss what developments are required in space-based quantum experiments to improve on the current measurement of the Schwarzschild radius of the Earth.