Estimating space-time parameters with a quantum probe in a lossy environment

We study the problem of estimating the Schwarzschild radius of a massive body using Gaussian quantum probe states. Previous calculations assumed that the probe state remained pure after propagating a large distance. In a realistic scenario, there would be inevitable losses. Here we introduce a practical approach to calculate the Quantum Fisher Informations (QFIs) for a quantum probe that has passed through a lossy channel. Whilst for many situations loss means coherent states are optimal, we identify certain situations for which squeezed states have an advantage. We also study the effect of the frequency profile of the wavepacket propagating from Alice to Bob. There exists an optimal operating point for a chosen mode profile. In particular, employing a smooth rectangular frequency profile significantly improves the error bound on the Schwarzschild radius compared to a Gaussian frequency profile.