  It is sometimes believed that small quantum gravity effects can encode
information as `delicate correlations' in Hawking radiation, thus saving
unitarity while maintaining a semiclassical horizon. A recently derived
inequality showed that this belief is incorrect: one must have order unity
corrections to low energy evolution at the horizon (i.e. fuzzballs) to remove
entanglement between radiation and the hole. In this paper we take several
models of `small corrections' and compute the entanglement entropy numerically;
in each case this entanglement is seen to monotonically grow, in agreement with
the general inequality. We also construct a model of `burning paper', where the
entanglement is found to rise and then return to zero, in agreement with the
general arguments of Page. We then note that the fuzzball structure of string
microstates offers a version of `complementarity'. Low energy evolution is
modified by order unity, resolving the information problem, while for high
energy infalling modes ($E>> kT$) we may be able to replace correlators by
their ensemble averaged values. Israel (and others) have suggested that this
ensemble sum can be represented in the thermo-field-dynamics language as an
entangled sum over two copies of the system, giving the two sides of the
extended black hole diagram. Thus high energy correlators in a microstate may
be approximated by correlators in a spacetime with horizons, with the ensemble
sum over microstates acting like the `sewing' prescription of conformal field
theory.
