Light escaping a massive body loses energy: its waves stretch, and its color shifts toward the red. This is gravitational redshift, known since the work of Einstein. But at the quantum level, a photon is not just a wave but a lump of energy that can be described by a set of parameters: for example, its initial phase and the 'weight' of its superposition (how much the particle is in two states at once). The problem is that any real communication channel or experimental setup introduces noise.
The authors considered two types of noise: amplitude damping (as if some photons are lost along the way) and dephasing (when random jolts disrupt the wave's synchronization). In both cases, the standard Cramér–Rao quantum limit, a familiar tool for estimating precision, fails for two parameters simultaneously. It paints too rosy a picture. So the researchers turned to stricter bounds: Holevo and Nagooka. Their calculation is no easy task, and here they had to use semidefinite programming (a method for solving complex systems of inequalities). It turned out that the Nagooka bound is the tightest, closest to reality.
When a third parameter enters the game, the strength of the gravitational redshift itself, the picture changes. With strong coupling to the noise channel, the precision of estimating all three parameters sharply increases. For dephasing, the 'memory' of the environment also matters: if the noise decays slowly (sub-Ohmic regime), measurements are more precise than with fast forgetting (super-Ohmic).
The Nagooka bound used here was proposed in 1989 but long remained in the shadow of simpler methods. Only with the development of computational power has it begun to be applied in practice.
In the novel 'Light of Distant Stars,' astronauts use quantum measurements of distorted light to determine how much space is curved around a black hole.