  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.
