Technical meaning of "no faster-than-light communication"? I often read sentences like "relativity forbid faster-than-light communication" or "quantum entanglement cannot be used to convey information faster than light", but it seems to me that the formalization of "faster-than-light communication is not possible" is harder than it might look.

Assume that me and a friend, the "skeptics", want to investigate if
Alice and Bob can transmit information faster than light, as they
claim they're able to do. Alice and I travel very far away, while my
friend and Bob stay together here, and each group reaches points in
space-time that are space-like separated. My friend has to record
everything Bob would have claimed to "receive" from Alice. Now, at the
last moment, I think of something, let's say, the first lines of a
poem, and ask Alice to send these sentences to Bob; then she touches
her head and looks concentrated, says "it's done", and we reach the
point in space-time where we have all four agreed to meet, and my
friend tells me that Bob "received" exactly the lines of the poem.

This story surely sounds extraordinary; but I don't see which rule of physics would forbid it to happen: would some force forbid Bob to articulate the precise words Alice was supposed to send him?
Moreover, if we repeat such experiments a certain number of times and Alice and Bob always succeed, I don't think I can gain more than a statistical proof of their power, but not an absolute one.
What do sentences like "such theory forbids faster-than-light communication" precisely mean?
In fact, the only definition I can think of is a counterfactual one: "Bob receives exactly the message Alice was supposed to send him, and it would have been the case even if I had asked Alice to send any other message". However, I am not comfortable with counterfactual thinking, and I don't really see how it makes things more formal.
 A: At least in the context of things like the no-communication theorem, the technical meaning of "no communication" is very clear:
Alice and Bob have coordinated to perform measurements on two halves of an entangled state at spacelike separation. Can Bob say anything about what Alice did? Note: Not just "Can Bob receive a specifically coded message Alice is trying to send him?", but whether he can say anything at all, with any degree of certainty.
The claim that "faster-than-light communication via entanglement is impossible" is precisely that the answer to this is "No." - Bob cannot perform any measurement that would allow him any sort of inference about what Alice did, neither a certain nor a statistical one. The entity that tells us what Bob's probabilities to measure certain results are is a density matrix $\rho_B$ that is obtained from the full Alice+Bob state $\rho_{AB}$ via a partial trace, and this partial trace is the same no matter what Alice does.
And sure, if you're looking to experimentally test this, you can always claim that only yields "statistical proof" (the test would have to be that Bob cannot, on average (an inherently statistical notion!), do better than random chance in trying to guess what Alice did) - maybe next time Bob can suddenly communicate with Alice and quantum mechanics as we understand it has stopped working. But this "objection" is an objection you can bring against any experimental test, whether it's testing faster-than-light communication or Newtonian gravity ("Hah, you're claiming rocks always fall down to earth but maybe you're just testing the wrong rocks! Your proof is just statistical!" is the more obviously misguided attack on Newtonian gravity), and sure, we need to make sure we're not just testing special cases, but that's really part of good experimental design in all situations. The reason we repeat experiments (sometimes millions and millions of times) is precisely to ensure we haven't just measured an outlier.
A: For simplicity, let's say that Alice herself only moves at non-relativistic velocities, relative to Bob. Bob records the time he sent the message, $t_0$. Alice records the distance she traveled away from Bob $D$, and the time at which she receives Bob's message, $t_1$. Then the absence of superluminal signaling implies that
\begin{equation}
\frac{D}{t_1 - t_0} \leq c
\end{equation}
where $c$ is the speed of light.
Note that correlations caused by quantum entanglement do not constitute a signal that can be used to transfer information, so it would not be valid to interpret $t_0$ and $t_1$ as the times that Bob and Alice make measurements on their halves of an entangled Bell pair, in the above formula.
A: Here is why the universe we happen to inhabit is not so constituted as to permit any sort of faster-than-light information transmission:
If it were so constituted, then depending on what one's state of motion was, it would be possible to experience effects before their causes had even happened. So for example, in our inertial frame of reference we would see someone hit the pavement while in another frame of reference, that someone had not yet jumped off the roof.
That kind of universe would be acausal and living in it would be extraordinarily difficult if not outright impossible.
