  Determining the ultimate precision limit for measurements on a sub-wavelength
particle with coherent laser light is a goal with applications in areas as
diverse as biophysics and nanotechnology. Here, we demonstrate that surrounding
such a particle with a complex scattering environment does, on average, not
have any influence on the mean quantum Fisher information associated with
measurements on the particle. As a remarkable consequence, the average
precision that can be achieved when estimating the particle's properties is the
same in the ballistic and in the diffusive scattering regime, independently of
the particle's position within its complex environment. This invariance law
breaks down only in the regime of Anderson localization, due to increased
speckle correlations. Finally, we show how these results connect to the mean
quantum Fisher information achievable with spatially optimized input fields.
