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.