(En)Tangled: Escape from Locality
When Rapunzel was stuck in her tower, she felt limited because she could only interact with and affect the state of what was in her immediate vicinity. Ultimately, she, like everyone else, was limited by the principle of locality, which states that an object is only influenced by its immediate surroundings. The main difference was that her limited motion meant she could not get close enough to as many people or things.
Even if Rapunzel had a phone, she would still be limited by locality. Although she could communicate with Flynn from a distance, this is only because her phone generates an electromagnetic disturbance that could propagate to Flynn’s. There would be no direct, local interaction between the phones but rather a chain of local interactions that would ultimately connect them. An important consequence of this limitation is that their communication cannot be instantaneous. The signal needs time to move from one phone to another because it travels at a finite speed.
This speed is the speed of light, which is the fastest that anything in our universe can travel. This fundamental limitation on how quickly two different points in space can communicate is a key tenet of special relativity and famously led Einstein to suggest that quantum mechanics is “an incomplete theory.” He objected to the disregard that quantum entanglement holds for his all-important principle.
Generally, chameleons are too large and needy to behave quantum mechanically, but a similar protocol can be used by physicists to entangle much smaller things, like photons or atoms.
Suppose that Rapunzel had a box containing two chameleons, one of which is alive. These chameleons are identical, so much so that they are impossible to tell apart. If the original box is split into two boxes, each containing one chameleon, in such a way that it is impossible to know which box contains the living and which contains the dead, then the two chameleons will be entangled. Based on the original knowledge of only one living chameleon and the law of preservation of chameleon lives, checking the health of the resident of one of the boxes will also inform the measurer of the other chameleon’s status.
The key difference between the classical and quantum worldview is that in the former, this occurs because the chameleon in the opened box was always dead or alive and therefore so was the other chameleon. Nothing physical changed, just the observer’s knowledge. In the latter view, each was in a quantum superposition of being dead and being alive.
When Rapunzel opens the box, this superposition collapses, meaning that there was a real, meaningful change in the state of the first chameleon. Therefore, the state of the second, unobserved chameleon must also collapse from a superposition to the state opposite to what the first chameleon was observed as. Even if the two chameleons were brought millions of light years apart, the second one’s superposition would collapse instantaneously when the first box was opened.
The fact that measuring the first also affected the state of the second is the definition of entanglement. Generally, chameleons are too large and needy to behave quantum mechanically, but a similar protocol can be used by physicists to entangle much smaller things, like photons or atoms. This is a violation of locality, just like Einstein said, but it has also been verified experimentally many times in the decades since he issued his original objection.
This all makes it very tempting to suggest that all Rapunzel really needed was a pair of quantum Pascals to overcome the limitations of her tower. The first problem with this thinking is that living things are too large, too boisterous, and too energy-dispersing to actually be in quantum mechanical superpositions, which is why weird quantum phenomena stay in the realm of atoms and molecules and do not intrude on our daily, macroscopic lives. Secondly, she would have no control over which state her chameleon, and therefore the other chameleon, would collapse into, so she could not send a definite message. Both would just get a random string of living and dead chameleons if they repeated the experiment over and over again. Rapunzel could use entanglement to escape locality, but even that has very definite limits.