From my admittedly layperson pop-sci level of understanding, that's easier than it seems: there's simply nothing you can do to one part of the entangled pair that will result in observable side-effects to the other.
> An important assumption going into the theorem is that neither Alice nor Bob is allowed, in any way, to affect the preparation of the initial state. If Alice were allowed to take part in the preparation of the initial state, it would be trivially easy for her to encode a message into it.
Couldn't the galactic emperor distribute a collection of entangled particles to a remote outpost, one for each day, that can be manipulated like a dead man's switch?
Every day, a light turns green, the emperor is alive?
The “has always” is in quotes because it’s a useful lie. You kind of need to really understand the double slit experiment to get quantum fields, superpositions, and how that related to entanglement. Took me years and years of occasional YouTube physics videos before it finally clicked. But if entanglement still doesn’t make sense, I’d start by trying to understand the double slit experiment. It sounds way less awesome than entanglement, but it isn’t really. Double slit is in fact awesome and just as weird. Entanglement is way less cool than it sounds, and no, not actually a way of cheating the speed of light limit for information transmission.
I still have zero concept of why it's impossible. Help?
Anyway, my layperson explanation is as follows. Let Alice and Bob share an entangled system, but are unable to communicate normally (e.g. are lightyears apart). A ice can measure her part of the system to instantaneously affect what Bob has. However, Bob doesn't know what result Alice obtained. So, in trying to figure out what Alice did to affect his system, he has to average out Alice's possible effects on his system i.e. the possible measurements that Alice could have obtained. This averaging procedure makes it so that Bob doesn't gain any information on what system he now has. In fact, he has exactly the same information about what possible measurements he would make after Alice measured her system than information he had before Alice measured her system.
Of course, if Alice was able to tell Bob the result of her measurement, then Bob wouldn't have to do any averaging and would gain information about his system. But that happening means that Alice is communicating her result to Bob classically, at slower-than-light speeds.