NIST Physicists Show Ion Pairs Perform Enhanced 'Spooky Action'

If that happens, the entanglement would be broken, and Charlie wouldn't know.

I was especially responding to this part.

This (intuitively) makes more sense to me than saying "information traveled" and "action at a distance"

You can not get away from non-locality in quantum mechanics. Because the outcome is only determined when a measurement is performed, this »information« has to travel to the entangled partner when the measurement is made. And this must happen instantaneously. In the classical example you can avoid this because the outcome is already determined during the setup.

I said that "there could be several different degrees of freedom to measure on" I meant this to exactly correspond to "the exact outcome is only established when one looks in the first box".

This are three different things. You can of course decide to measure the spin along any axis you like, that is you can choose the observable. When you do this, then you are not measuring different degrees of freedom, you are measuring the same degree of freedom, the spin, in different bases. But, and that is what I meant, even if you always measure the spin along the same axis, say along the positive x axis, then the spin of an electron in a superposition of spin either +1 or -1 along this axis is only determined at the time of the measurement. Before that measurement the electron had no definite spin along this axis.

What you're describing is a local hidden variable theory. In this case, the "hidden variable" is who has which fruit. It's "local" because there's no remote influence -- Bob eating his fruit doesn't change which one Alice has.

Bell's theorem places limits on the correlations that a theory like this can explain. Here's a simple example of a Bell violation:

1) Alice and Bob get together and conspire. They can do anything they want here.

2) Alice and Bob are then taken to separate interrogation chambers, to be quizzed by separate interrogators.

3) Each interrogator randomly (and independently) chooses one of three yes-or-no questions to ask their subject. Both interrogators are working from the same list of 3 questions. Alice and Bob are asked the questions and the questions and their answers are recorded.

4) This whole process is repeated many times. When the results are tabulated, it's found that every time Alice and Bob were asked the same question, they gave the same answer. But when they were asked different questions, 75% of the time they gave different answers.

How could Alice and Bob have achieved this? Classically, they couldn't. In order to always give the same answer to the same question, they'd have to (either directly or indirectly) agree beforehand on which answer to give to which question.

Since there are three possible questions but only two possible answers to them, any strategy they pick must give the same answer to (at least) two of the three questions. Therefore, if their interrogators pick different questions there's at least a 1/3 chance they'll pick the questions with the same answer.

But with quantum entanglement, the chance of getting the same answer from both is can be only 1/4. Something weirder is going on.

(BTW, for completeness, here's how the quantum results I described can be achieved: Alice and Bob are replaced by electrons, prepared in the (entangled) singlet state. The "interrogations" are measurements of their spin (up or down) on the 0°, 120°, or 240° axes, with one of the detectors set up upside-down. This is not the way it's actually done, but it's easier to explain & see the paradox this way.)

One has to be careful what one means here. I meant it in the following specific sense and we have good experimental evidence for this.

[...] there is no time, no matter how far the spins are separated, in which you could measure the second spin to have the same orientation as the one you just measured [...]

If an interpretation makes use of non-locality to explain this fact is a different matter, but experimentally it looks exactly like measurements instantaneously affecting entangled partners.

Oh cool! This experiment is very fun, not just from the results side. I was able to get a tour with the photonic side of this set-up (not the atomic side, as this paper elucidates). The photons are made entangled, then split and sent into some fiber-optic cables that run about in the tracts in the hallways. The rooms used to measure the entanglements are sufficiently far apart, but due to budgets, are at right angles from the source. We only went to one of the rooms, but the measurement devices are housed in marijuana grow pods you can buy, as they tend to keep the temperature stable, have access for the wires and cryogenic tubes, will keep out/in EM noise, and are pretty cheap. There was a line on the floor of that room, off in the corner, that had where the 'light speed' signal from the other measurement room stopped when the measurement (in the room we were in) was made, proving that the measurements could not interact. If you ever get a chance, get a tour at NIST, it is well worth whatever strings you have to pull. Getting to see The United States Second (where the US measures all of out time from) was a real highlight of my scientific career.