Information Theoretical Approach to Detecting Quantum Gravitational Corrections

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.