This is my first post in Physics Overflow so if this question doesn't fit here please let me know.

In the paper On the need for soft dressing the authors compare two approaches to IR divergences in QED: the Bloch-Nordsieck approach of computing only inclusive cross sections tracing out soft photons bellow one IR cutoff and the Faddeev-Kulish approach of dressing asymptotic particles with soft photon clouds in coherent states, thereby obtaining one IR finite $\mathcal{S}$-matrix. Their comparison is in the situation on which the incoming state is either a finite superposition or a wavepacket and conclude in favor of the Faddeev-Kulish approach for this scenario. Now then, in the conclusions they state the following conjecture about black holes:

Our results may have implications for the black hole information loss problem. Virtually all discussions of information loss in the black hole context rely on the possibility of localizing particles – from throwing a particle into a black hole to keeping information localized. We argued above that normalizable (and in particular localized) states are necessarily accompanied by soft radiation. It is well known that the absorption cross-section of radiation with frequency $\omega$ vanishes as $\omega\to 0$ and therefore it seems plausible that, whenever a localized particle is thrown into a black hole, **the soft part of its state which is strongly correlated with the hard part remains outside the black hole**. If this is true a black hole geometry is always in a mixed state which is purified by radiation outside the horizon.

I'm interested in this conjecture and in particular how it could be made precise. For that matter I'm trying to consider a very basic setup to understand properly what happens to a soft particle heading into a black hole. My objective is to understand the description of "the soft part of its state remaining outside of the horizon", i.e., the state after the reflection by the black hole horizon.

So I considered as a first step a massless Klein-Gordon field propagating on a maximally exended Schwarzschild background, since it seems easier for a first computation than the dynamical collapse to a black hole. At $\mathcal{I}^\pm$ there is no ambiguity in the definition of positive frequency and we know how to define in/out particle states: we basically quantize the field demanding a decomposition into positive/negative frequency with respect to advanced/retarded times. On the other hand at $\mathcal{H}^\pm$ the future and past horizons there is ambiguity. The problem then would be to take one initial state defined on a quantization on ${\Sigma}^- = \mathcal{I}^-\cup \mathcal{H}^-$ and expand it via Bogolubov coefficients on the appropriate basis for a quantization defined on $\Sigma^+ = \mathcal{I}^+\cup \mathcal{H}^+$.

The initial state to capture the authors' proposal is clearly one soft incoming particle on $\mathcal{I}^-$, i.e., with frequency $\omega\to 0$. Since on $\mathcal{I}^-$ we know the appropriate quantization this seems to be of no problem.

Now on $\mathcal{I}^+$ we also know the appropriate quantization: we pick basis modes which are zero on the future horizon and positive frequency with respect to retarded time $u$.

But now I believe I need to pick modes on $\mathcal{H}^+$, the future horizon, describing what the authors are calling "the soft part of the state remaining outside of the black hole". But here there is one issue: since there is no unambiguous notion of positive frequency on the horizon, which should we pick to achieve the authors proposal?

So my question here is: what are the modes describing "the soft part of the state remaining outside of the black hole"? Am I on the righ track that these should be modes with some Cauchy data on $\mathcal{H}^+$ and zero on $\mathcal{I}^+$? How does pick these modes after all?