Although mini black holes are still theoretical, if one were to be created using the LHC, would it be possible to monitor the events inside of this black hole using an entangled photon as a probe. The properties of the entangled photon in our space should mirror those of its twin inside of the black hole. If the entangled photon is destroyed, wouldn't its mirror outside of the black hole also cease to exist?
If the entangled photon is destroyed, wouldn't its mirror outside of the black hole also cease to exist?
Nope. Entanglement doesn't work like that.
You're expecting your right sock to catch fire just because you threw your left sock into a fire. You're violating the no-communication theorem.
What you could do (in principle, not in practice, and perhaps not even in principle depending on who's right about what happens inside black holes) is drop the entangled photon in, capture all of the Hawking radiation that comes off of the black hole until it decays, do a lot (a lot) of quantum computation to undo all the mixing that happened, thereby recover the entangled partner (maybe), then do some kind of interesting comparison.
I would tend to say no. The reason is that while you can send a part of an entangled pair into a black hole the component states actually do not exist, and secondly from the perspective of the outside the other part of the pair that enters the black hole never in fact crosses the horizon.
Quantum entanglement is a case where the total quantum state is known and the component states are not. In fact not only do you not know the component states, they actually in fact have no ontological meaning or existence. An entangled pair has two qubits of information, and if the actual internal states existed in some ontological sense this would mean there would in fact be more information. This additional information would correspond to a hidden variable, and if "knowable" as information would then be local. This violates Bell's theorem, for such as system would obey the Bell inequalities.
If you are throwing an part of the entangled pair to the black hole its coordinate position given by the tortoise or delay coordinate $$ r^*~=~r~-~2m~ln(r~-~2m), $$ for $m~=~GM/c^2$. This diverges as $r~\rightarrow~2m$ at the event horizon. This means it would take an infinite time $t~=~r^*/c$ for anything to reach the black hole.
The idea is not entirely crazy though. If the black hole is itself quantum mechanical the event horizon has an uncertainty in its position. This will mean an uncertainty will exist whether quantum states are exterior or interior to a black hole. It will of course not be possible to perform quantum black hole experiments. However, a graviton is equivalent to an entangled or bound state of two gluons in a net colorless state. The S-dual of the di-gluon, which is strongly interacting, is the graviton that is very weakly interacting. This means with heavy ion experiments, such as Pb ions and the ALICE detector, could perform experiments on S-dual black holes, which would be quark gluon plasmas.
A few points to consider:
We don't know how entanglement works. Either inside a black hole or outside.
As Craig states in his answer, during measurement of an entangled quantum state, it is not possible for one observer, by making a measurement of a subsystem of the total state, to communicate information to another observer.
Entanglement is extremely sensitive to any and all influences and disentanglement is extremely likely.
We can't / haven't made a mini black hole at the LHC
If mini black holes are created by the LHC, they will not last long enough to send a probe into them. (Hopefully, otherwise we may have a problem)
Destroying one photon does not imply the other photon will be destroyed.
The properties of the entangled photon in our space should mirror those of its twin inside of the black hole.
How do we know that is true as, by definition, we have no idea what is going on on the other side of the black hole.?
@Craig Gidney has a good description of what you could do with micro or quantum BHs. If you can detect all the Hawking radiation coming out before and as it evaporates completely, you can try doing an accounting of the different quantum numbers come out. As he says with a lot of still not very clear computation (you might have tens or hundreds or thousands of particles come out, I have not done or seen the computation for the BH masses that might be produced).
But once one starts getting some of them they might be reproducible statistically, and analyzable. And then one can start having entangled photons or other particles interacts with the BH and some will get absorbed (or get to within a Planck length of the horizon, maybe for practical purposes it's the same and if not it'd be interesting to 'see'), and then one can determine if the Hawking radiation coming out preserves its information. For a photon the conserved quantum properties in a BH is energy, zero charge, and spin, per the no-hair theorem. The question could then get answered as to whether those and the other no-hair quantum numbers of the photon like parity and charge parity get conserved.
Once micro BHs start getting produced, if ever, in enough quantity that we can experiment with them, other kinds of tests of the conservation of information (quantum numbers) would be tried.
Richard, apart from entanglement your question opened up other interesting possibilities. +1 for that.