How to minimize the decoherence caused by black holes

We consider an experimentalist, Alice, who creates a quantum superposition of a charged or massive body outside of a black hole (or, more generally, in the presence of a Killing horizon). It was previously shown that emission of soft photons/gravitons into the black hole will result in the decoherence of the components of the superposition if it is held open for a sufficiently long span of time. However, at any finite time, tc, during the process, it is not obvious how much decoherence has irrevocably occurred. Equivalently, it is not obvious how much information an observer inside the black hole can extract about Alice's superposition prior to time tc. In this paper, we solve for the optimal experimental protocol to be followed by Alice for t>tc so as to minimize the decoherence of the components of her superposition. More precisely, given the entangling radiation that has passed through the horizon prior to the cross-section C corresponding to the time t=tc in Alice's lab, we determine the "optimal purification" of this radiation beyond C such that the global quantum state of the radiation through the horizon has maximal overlap (quantum fidelity) with the Hartle-Hawking or Unruh vacuum. Due to the intricate low frequency entanglement structure of the quantum field theory vacuum state, we find this optimal purification to be nontrivial. In particular, even if Alice has already "closed" her superposition by bringing the components back together, we find that she can decrease the net decoherence of the components of her superposition somewhat by reopening it and performing further manipulations.