The black hole information puzzle and the quantum de Finetti theorem

The black hole information puzzle arises from a discrepancy between conclusions drawn from general relativity and quantum theory about the nature of the radiation emitted by black holes. According to Hawking's original argument, the radiation is thermal and its entropy thus increases monotonously as the black hole evaporates. Conversely, due to the reversibility of time evolution according to quantum theory, the radiation entropy should decrease in the final stages of evaporation, as predicted by the Page curve. This behaviour has been confirmed by new calculations based on the replica trick, which also exhibited its geometrical origin: spacetime wormholes that form between the replicas. Here 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 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 its physical meaning in a many-black-holes scenario. Our analysis hints at an interpretation of the replica wormholes as the geometrical representation of the correlation between the black holes, which is mediated by $W$. It also suggests an extension of the widely used random unitary model of black holes, which we support with some new non-trivial checks.