  A black hole evaporates by Hawking radiation. Each mode of that radiation is
thermal. If the total state is nevertheless to be pure, modes must be
entangled. Estimating the minimum size of this entanglement has been an
important outstanding issue. We develop a new theory of constrained random
symplectic transformations, based on that the total state is pure and Gaussian
with given marginals. In the random constrained symplectic model we then
compute the distribution of mode-mode correlations, from which we bound
mode-mode entanglement. Modes of frequency much larger than $\frac{k_B
T_{H}(t)}{\hbar}$ are not populated at time $t$ and drop out of the
analysis.Among the other modes we find that correlations and hence entanglement
between relatively thinly populated modes (early-time high-frequency modes
and/or late modes of any frequency) to be strongly suppressed. Relatively
highly populated modes (early-time low-frequency modes) can on the other hand
be strongly correlated, but a detailed analysis reveals that they are
nevertheless very unlikely to be entangled. Our analysis hence establishes that
restoring unitarity after a complete evaporation of a black hole does not
require any significant quantum entanglement between any pair of Hawking modes.
Our analysis further gives exact general expressions for the distribution of
mode-mode correlations in random, pure, Gaussian states with given marginals,
which may have applications beyond black hole physics.
