1591: Bell's Theorem
Title text: The no-communication theorem states that no communication about the no-communication theorem can clear up the misunderstanding quickly enough to allow faster-than-light signaling.
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'Global hidden variables' are another story: if there is classical information shared across systems (perhaps by superliminal communication) even up to superdeterminism where the universe is just reading off a script, any correlations can be explained away. But this is unsatisfying.
The prefered resolution of the paradox is to not insist (as early physicists did) that the universe's state is a collection of bits (classical information), but treat it as a collection of qubits (quantum information).
Ponytail begins reading Bell's theorem to Cueball, who is standing 5 meters away. Cueball responds with a misunderstanding of Bell's Theorem in 1 nanosecond. The speed of light in a vacuum is 299,792,458 meters per second. In one nanosecond, the light from Ponytail would only have traveled 0.299 meters, thus Cueball misunderstands Bell's Theorem faster than the light from Ponytail reading the Theorem can reach him, which implies that faster-than-light communication occurred to set up the misunderstanding.
In quantum mechanics (QM), 'measurement' is the process of allowing a small system to interact with its environment in a controlled way. The interaction allows information about the system's state to escape to the environment, producing an 'observation'. If the measurement apparatus is governed by classical mechanics (impossible in reality, but a very common simplification for the purposes of calculation), then the observation can be thought of as classical information, a bit (yes/no answer) in the simplest case. While the system may have been in any one of infinitely many states before the measurement (each a superposition of classical states), the fact that the measurement must leave it consistent with the classical result means that it can end up in only finitely many states afterwards. This is the 'wave-function collapse' of early QM, popularized by Schrodinger's cat, but unrelated to the Heisenberg uncertainty principle, which lay audiences often confuse it with.
Modern quantum mechanics acknowledges that the environment is not classical, and that wave-function collapse happens by a (comparatively) gradual process called 'decoherence', where information leaving the system is made up for by information coming from the environment that drives the system closer and closer to one of the finitely many state predicted by the simplified model above. (I.e., if a "Schrodinger's cat" is in a half-and-half superposition of the states "dead" and "alive", when its liveness is measured, the ratios of "dead" and "alive" will shift rapidly towards (though not quite reach) 0 and 100% or 100 and 0%. For all but the shortest time scales, the cat's post-measurement state might as well be classical.)
Entanglement is a situation where the future outcomes of two or more measurements that would be independent in a classical world are nonetheless correlated. For example, two widely separated electrons could be in a state where, considered individually, each is in a superimposed spin-up/spin-down state, but if one is measured as spin-up, the others will necessarily be measured as spin-down. This is untroubling if the two electrons are modeled as a single system, but strange-seeming if we think of them as separate: how did the measurement of the first electron allow information from the environment around it affect the far-away second electron? It seems like the electrons are communicating, potentially at superliminal speeds, which would violate either relativity or causality. (In actuality, there's a fairly simple proof that correlations from entanglement can't be used to communicate, and causality and relativity are safe. But that doesn't make the seemingly faster-than-light effects much less of a surprise.)
One can try to address these concerns by considering 'local hidden variables', classical properties of a local system (like a single electron) that could have been observed but were not. For example, perhaps a classical part of the electrons' state lets them "agree" on a future classical state at the moment the are entangled, and then they just reveal that state in the future. But this becomes unwieldy: there are infinitely many possible future observations the electrons would have to agree on, and it seems difficult to do this without infinitely many local hidden variables.
The title text, with 4 negatives, tries to be as confusing as possible. The real No-Communication Theorem states that although determination of the state of one half of an entangled pair immediately determines that of the other half, however far away it may be, there's no way for the observer of the other half to see if he's the first to find out the state or whether it'd already been determined by the first observer. Thus, no information travels from one observer to the other. Randall's version is recursive. It hypothesises a method of communication whereby somebody misunderstanding the no-communication theorem (which also happens faster than the speed of light) could function as the reception of a faster-than-light signal. However, it goes on to point out that turning the signal off requires clearing up the confusion which takes much, much longer, thus neatly restoring the normality of slower than light communication.
[First frame captioned: t = 0 nanoseconds]
Ponytail, holding a piece of paper and facing to the right: This is called Bell's Theorem. It was first–
[A double-headed arrow links the characters in the two frames. The arrow is labelled "5 meters".]
[Second frame captioned: t = 1 nanosecond]
Cueball, facing to the left towards Ponytail: Wow, faster-than-light communication is possible!
Caption: Bell's Second Theorem: Misunderstandings of Bell's Theorem happen so fast that they violate locality.
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