Demystifying the replica trick calculation of the black hole radiation entropy

The Page curve describing the radiation entropy of a unitarily evaporating black hole has recently been obtained by new calculations based on the replica trick. We analyse the discrepancy between these and Hawking's original conclusions from a quantum information theory viewpoint, using in particular the quantum de Finetti theorem. The theorem implies the existence of extra information, W, which is neither part of the black hole nor the radiation, but plays the role of a reference. The entropy obtained via the replica trick can then be identified to be the entropy S(R|W) of the radiation conditioned on the reference W, whereas Hawking's original result corresponds to the non-conditional entropy S(R). The entropy S(R|W), which mathematically is an ensemble average, gains an operational meaning in an experiment with N independently prepared black holes: for large N, it equals the regularized entropy of their joint radiation, S(R_1…R_N)/N. The discrepancy between this entropy and S(R) implies that the black holes are correlated, that is geometrically captured by the replica wormholes. In total, I will give three different interpretations of the radiation entropy calculated via the replica trick. Furthermore, I will briefly discuss the implications of ensemble interpretation in light of free probability theory, which offers the tools to deal with the effect of replica symmetry breaking in a refined calculation of the radiation entropy. (Based on the joint work (https://arxiv.org/abs/2110.14653) with Renato Renner.)

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