Entropy is Conserved in Hawking Radiation as Tunneling: a Revisit of the Black Hole Information Loss Paradox

We revisit in detail the paradox of black hole information loss due to Hawking radiation as tunneling. We compute the amount of information encoded in correlations among Hawking radiations for a variety of black holes, including the Schwarzchild black hole, the Reissner-Nordstr\"{o}m black hole, the Kerr black hole, and the Kerr-Newman black hole. The special case of tunneling through a quantum horizon is also considered. Within a phenomenological treatment based on the accepted emission probability spectrum from a black hole, we find that information is leaked out hidden in the correlations of Hawking radiation. The recovery of this previously unaccounted for information helps to conserve the total entropy of a system composed of a black hole plus its radiations. We thus conclude, irrespective of the microscopic picture for black hole collapsing, the associated radiation process: Hawking radiation as tunneling, is consistent with unitarity as required by quantum mechanics.