The Equivalence Principle necessitates a mixed state description for the exterior of a black hole? I am going through this review paper on the black hole information paradox. In section 2.2 page 10, the author argues the following:

One of the basic tenet of general relativity is the Equivalence
principle, which states that the spacetime near the horizon can be viewed locally as Minkowski spacetime. Therefore very close to horizon the Rindler description of the spacetime can be adopted. Therefore the horizon hides correlations in the same fashion as a Rindler horizon does. Unless the correlations across the horizon are turned off, the description on the outside will remain mixed. Thus, the Equivalence Principle also necessitates a mixed state description
for the exterior

I understand that the equivalence principle tells us that spacetime near the (BH) horizon can be treated as locally Minkowski. It can also be shown mathematically that this manifests as the Rindler description of the metric near the horizon.
I however then fail to understand the author in the subsequent lines where he claims that the horizon hides "correlations" in the same way as the Rindler horizon does. What correlations is the author referring to? Is it the correlations between the outgoing radiation and the interior? But doesn't the no-hiding theorem precisely lead to the result that information cannot be hidden in correlations between the interior and exterior? How exactly does equivalence principle necessitate a mixed state description here?
 A: This answer draws some inspirations from Sec. II of arXiv: 1703.02140, which is also a review on information loss in black holes.
The author seems to refer to the fact that there is entanglement between the inside and outside of the horizon, and an outside observer has no access to the modes inside the horizon. For example, for a free scalar field in Minkowski spacetime one has, to leading order,
$$\langle\Psi\vert\hat{\phi}(x_1)\hat{\phi}(x_2)\vert\Psi\rangle \sim \frac{U(x_1, x_2)}{\sigma(x_1,x_2)}, \tag{5}$$
which corresponds to Eq. (5) on the paper I mentioned. $U$ is some smooth function, while $\sigma(x_1,x_2)$ is the squared geodesic distance between $x_1$ and $x_2$. You see then that the field bears correlations between different points in spacetime. Notice that the expression diverges as $x_1 \to x_2$. On the other hand,
$$\langle\Psi\vert\hat{\phi}(x_1)\vert\Psi\rangle\langle\Psi\vert\hat{\phi}(x_2)\vert\Psi\rangle \to [\langle\Psi\vert\hat{\phi}(x_2)\vert\Psi\rangle]^2 \tag{6}$$
as $x_1 \to x_2$ (this corresponds to Eq. (6) on the paper I mentioned).
Since Eq. (5) diverges in the $x_1 \to x_2$ limit and Eq. (6) doesn't, we see  there's entanglement between the regions around $x_1$ and $x_2$, which could be, for example, points on either side of the Rindler horizon. Hence, in this sense, the horizon hides a correlation. Once we trace out the inside of the horizon, we'll be left with a mixed state as a consequence.
The point of the author seems to be that, due to the Equivalence Principle, this very same thing will also apply to a black hole's event horizon.
