The black hole information puzzle and the quantum de Finetti theorem

Abstract

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 a black hole. According to Hawking's original argument, the radiation is thermal and its entropy thus increases monotonically as the black hole evaporates. Conversely, due to the reversibility of time evolution according to quantum theory, the radiation entropy should start to decrease after a certain time, as predicted by the Page curve. This decrease has been confirmed by new calculations based on the replica trick, which also exhibit 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 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 normalised entropy of their joint radiation, $S(R_1\cdots R_N)/N$. The discrepancy between this entropy and $S(R)$ implies that the black holes are correlated. The replica wormholes may thus be interpreted as the geometrical representation of this correlation. Our results also suggest a many-black-hole extension of the widely used random unitary model, which we support with non-trivial checks.

39+6 pages, 6 figures. v2: Largely rewritten to improve presentation; statement of the de Finetti theorem reformulated with a quantitative error corrected; technical proofs provided in the appendices; Section 4 expanded with the name "Missing Reference" changed to "Elusive Reference"

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• Black holes Entangled by Radiation