  In 1974 Steven Hawking showed that black holes emit thermal radiation, which
eventually causes them to evaporate. The problem of the fate of information in
this process is known as the "black hole information paradox". It inspired a
plethora of theoretical models which, for the most part, assume either a
fundamental loss of information or some form of quantum gravity. At variance to
the main trends, a conservative approach assuming information retrieval in
quantum correlation between Hawking particles was proposed and recently
developed within qubit toy-models. Here we leverage modern quantum information
to incarnate this idea in a realistic model of quantised radiation. To this end
we employ the formalism of quantum Gaussian states together with the
continuous-variables version of the quantum marginal problem. Using a rigorous
solution to the latter we show that the thermality of all Hawking particles is
consistent with a global pure state of the radiation. Surprisingly, we find out
that the radiation of an astrophysical black hole can be thermal until the very
last particle. In contrast, the thermality of Hawking radiation originating
from a microscopic black hole -- which is expected to evaporate through several
quanta -- is not excluded, though there are constraints on modes' frequencies.
Our result support the conservative resolution to the black hole information
paradox. Furthermore, it suggests a systematic programme for probing the global
state of Hawking radiation.
