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I don't know much about quantum teleportation but I read that scientists have succeeded in various quantum teleportation experiments. I'm wondering if one researcher were just inside the event horizon of a black hole, and another just outside, could the one outside teleport anything to the one inside? I'm assuming he'd never be able to know if he succeeded or not, and that the one inside could not teleport anything out.

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    $\begingroup$ Attention: Scientists haven't succeeded in teleportation (I'd better write an answer, though). So you are actually asking two different questions... $\endgroup$ – Helen - down with PCorrectness Apr 1 '17 at 11:44
  • $\begingroup$ sciencemag.org/news/2015/12/… $\endgroup$ – user126422 Apr 1 '17 at 17:27
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The answer is no, because quantum teleportation requires normal communication to work. The process is that we assume one experimenter has two qubits (small quantum systems), A and B, while another has a third, C. We assume B and C are initially in an entangled state. To perform teleportation, you measure A and B together, and then you call your friend and tell them what result you got. This then allows them to perform a procedure on C that recovers whatever information you had stored in A.

This works/is useful because it allows you to send A without measuring it and disturbing the information, but it does not allow you to send information unless you can also contact your friend classically to tell them what your measurement result was. So if one of you is inside a black hole, you can't perform the teleportation protocol.

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A clarification about the premises of this question: So far, scientists haven't succeeded in any teleportation experiments (and so far there is no hint that in principle teleportation is possible):

Up to day, what are sometimes called "teleportation experiments" involve the observation of entangled pairs of particles. Here we talk about "teleportation" of information, not of matter, but even this doesn't happen.

Such experiments measure a particle's attribute, e.g. spin, and infer the spin of the other particle of the pair, which is found at a large distance; the crucial point is that neither of the spins was determined before the measurement. So, by acting on the nearby particle, the far-away particle acquired its spin value. However, the experimenter neither chose nor modified the nearby spin. So, he didn't pass any information to the far-away particle. It's still a tremendous find in terms of quantum mechanics but certainly not teleportation...

Giving a short answer though to your hypothetical question, it will probably depend on what the means of teleportation will be ... and at which point inside a black hole information gets lost :\

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    $\begingroup$ This is not true. Consider this experiment (from 2009), where teleportation is accomplished after preparing the teleported qubit in a known state, which is then recovered afterwards arxiv.org/abs/0907.5240 $\endgroup$ – zeldredge Apr 1 '17 at 12:38
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    $\begingroup$ Or this one, in which the qubits were photon qubits distributed over a fiber network in Calgary arxiv.org/abs/1605.08814 $\endgroup$ – zeldredge Apr 1 '17 at 12:40
  • $\begingroup$ Okay, I had in mind Bell / Aspect type experiments. I'll read your links. $\endgroup$ – Helen - down with PCorrectness Apr 1 '17 at 12:43
  • $\begingroup$ After reading the links: 0907.5240 is about quantum cryptography. Teleportation here refers to transfering information securely, by issuing a key that enables the receiver to read his copy of the "message". It involves that "the result of the measurement on ion A is relayed through a classical communication channel", i.e. the receiving end must learn the result of a measurement at the sending end. Of course this means that for teleportation of information and matter there are no advantages over e.g. internet or remote 3-d printing (however there are notable ones for cryptography). $\endgroup$ – Helen - down with PCorrectness Apr 1 '17 at 16:36
  • $\begingroup$ @zeldredge About 1605.08814, I still try to wrap my brain around it, but I think it's of the same kind. And, this article popularizing the paper says the following: "The initial Bell measurement is made by Alice on the photon to be cloned and also on one photon in a pair of entangled photons. The result of this measurement is then sent to Bob, who also receives the other photon in the entangled pair. Using Alice's measurement information, Bob then makes a further measurement on the entangled photon, which puts it into the same quantum state as the cloned photon." $\endgroup$ – Helen - down with PCorrectness Apr 1 '17 at 16:38
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As stated in previous answers, and in the comments, quantum teleportation protocol let us to send the state of entangled qubits with the help of sub-luminal communication. In this video the essential concept is explained in a very friendly way. However the important part of the sub-luminal communication is not evplained in detail.

Schemattically we can perform the teleportation protocol as follows:

  • Taking the state $\phi_1$ we want to teleport and a couple of entangled systems $\psi_2$ and $\psi_3$, for short $\psi_{23}$.
  • Entangle the firs state with one of the system in $\psi_{12}$, obtaining a "very" entangled system $\psi_{123}$
  • Perform a series of transformations, which will lead to a superposition of the system $\psi_{12}$ and the third part $\psi_3$
  • By measuring and informing about the results in $\psi_{12}$ one can obtain the desired state copied in $\psi_3$.

Notice is important the fact that we have to inform about the state of $\psi_{12}$ in order to select correctly the state of $\psi_3$.

But there are technical problems: first of all, entanglement is a very fragile property, so the experiment must be taken carefully under controlled circumstances. Furthermore, this protoc ol works well (and was developed) in the context of quantum mechanichs, which treats time differently as Special/General Relativity.

Now let's move on to a black hole: near the event horizon, spacetime is perfectly regular (a free-falling observer cannot notice the difference between inside and outside the horizon), but the characteristic of the time coordinate changes. To make it simpler, let's think about Schwarschild BH, which is one of the simplest. In this spacetime, the time flow (the killing vector associated with time translations) goes form time-like to space-like once you cross the horizon, which means that once you entered a black hole, the only causal curve you can follow (the only trajectory permissible) goes trhough the singularity. Since thw communication used in the teleportation protocol is subluminal (and, therfor, causal), no information can escape of the black hole, so the teleportation protocol cannot hold here, as stated in the previous answers.

Allowing this caveat, it is possible that even the concept of information of a particle inside the black hole has no physical meaning. Since the qubits are respectively inside and outside of the horizon, it is not sure that the entanglement holds. This is related, if not the central point, of loss information paradox. Nowadays is thought that the information of a particle falling into a black hole is not contained inside the black hole, but that leaves an imprint in the horizon (c.f Soft Hair on Black Holes by S. Hawking et all.)

In 2015 an analogue with dumb holes was performed and looks like the entanglement could survive but in presence of firewalls for example, is not that easy (c.f. journals.aps.org/pra/abstract/10.1103/PhysRevA.81.032320 and journals.aps.org/prl/abstract/10.1103/PhysRevLett.115.031301‌​). So in conclusion: even if the teleportation protocol holds, if communication of states between inside and outside of black hole is possible, it is not sure that the basis of the transportation protocol (i.e. the quantum entanglement) can even exist.

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