An entangled pair of particles is created with one particle embedded onto the surface of a singularity. The other particle at some stated epsilon distance above the event horizon of the singularity. Can the property of entanglement be preserved or is decoherence immediate? If the quantum system is fully described by the eigenfunction thea what tests, i.e. operators, can be performed to prove entanglement is invariant under the operation of being embedded into the event horizon? Is there a Hermitian Operator which maps from outside to onto the surface of the singularity such that we can measure any of the systems quantum properties?
I am not going to address directly the issue of the singularity. This is in part because we really are not sure what that is exactly. It is a region where the Weyl curvature tensor corresponding to tidal forces diverges. This is at least in classical gravitation. Quantum mechanically it may mean something else, such as a topological charge associated with a sort of monodromy. I will however discuss the issue of entanglement and black holes. This is a bit of a serious problem.
There is this problem with how gravitation and quantum mechanics merge or function in a single system. It is often said we understand nothing of quantum gravity, and this is not quite so. Even with the based canonical quantization of gravity from the 1970s in a weak limit is computable and tells you something. This theoretical understanding is very limited and big open questions remain. Of course since then far more progress has been made. The AdS/CFT correspondence, the Raamsdonk equivalence between entanglement and spacetime and the RT formula are some of the more recent developments. These indicate how spacetime physics has a correspondence or maybe equivalency with quantum mechanics or quantum Yang-Mills fields. However, an obstruction exists that appears very stubborn.
The vacuum is filled with virtual pairs of fields. With a black hole the gravity field causes one of these pairs to fall into the black hole and the other to escape. This means the quantum particle or photon that escapes as Hawking radiation is entangled with the pair that falls into the black hole, and so this means Hawking radiation is entangled with the black hole. So at first blush there seems to be no problem. However, if we think of a thermal cavity heated to high temperature photons that escape are entangled with quantum states of atoms composing the cavity. Once the entanglement entropy reaches a maximum at half the energy released the subsequent photons released are entangled with prior photons released. This would hold with black holes as well, but because of the virtual pair nature of this radiation it means Hawking radiation previously emitted in a bipartite entanglement are now entangled not just with the black hole, but with more recently emitted radiation as well. This means a bipartite entanglement is transformed into a tripartite entanglement. Such transformations are not permitted by quantum unitary evolution. This is called quantum monogamy requirement, and what this suggests is unitarity fails. To prevent the failure of quantum mechanics some proposed a firewall that violates the equivalency principle. This is called a firewall. The fire wall proposal can be read here.
The firewall occurs when half the possible radiation is emitted, which is also the Page time. This also corresponds to the failure of a quantum error correction code. Error correction codes involve some deep mathematics; it is connected with the RT formula and I illustrate in my essay the connection with Mirzakhani's mathematics on the geodesics in hyperbolic spaces. Error correction is also tied with the packing of spheres or how oranges stack at the grocery store, the Kepler problem. This gets into the guts of what my paper is about. However focusing in an error correction corrects the mixing of information. Think of a library, in particular an elementary school library with little kids, and the patrons scramble up the order of books. The distance a books ends up from its right position is the Hamming distance. As the library gets mixed up an algorithm can manage this disordering. However, at about half mixing up things break down. The librarian has to virtually start over.