  One way to test quantum gravitational corrections is through black hole
physics. In this paper, We investigate the scales at which quantum
gravitational corrections can be detected in a black hole using information
theory. This is done by calculating the Kullback-Leibler divergence for the
probability distributions obtained from the Parikh-Wilczek formalism. We
observe that the quantum gravitational corrections increase the
Kullback-Leibler divergence as the mass of the black hole decreases, which is
expected as quantum gravitational corrections can be neglected for larger black
holes. However, we further observe that after a certain critical value, quantum
gravitational corrections tend to decrease again as the mass of the black hole
decreases. To understand the reason behind this behavior, we explicitly obtain
Fisher information about such quantum gravitational corrections and find that
it also increases as the mass decreases, but again, after a critical value, it
decreases. This is because at such a scale, quantum fluctuations dominate the
system and we lose information about the system. We obtain these results for
higher-dimensional black holes and observe this behavior for Kullback-Leibler
divergence and Fisher information depending on the dimensions of the black
hole. These results can quantify the scale dependence and dimension dependence
of the difficulty in detecting quantum gravitational corrections.
