So this question is about the strange phenomenon of quantum entanglement, or "spooky action at a distance", as Einstein called it. In particular, it's about the simplest conceivable case of quantum entanglement that you'll probably ever see, namely the one with two particles that are entangled in such a way that if one particle is observed to have spin up, then the other will have spin down, and vice versa.

So the story usually goes something like this: imagine that you have two particles that first interact in such a way that their combined spin wave function becomes a superposition of one having spin up and the other spin down, and vice versa. Then separate the two particles by some arbitrary distance. Now, if you measure the spin of particle A, say, and it turned out to be spin up, then *simultaneously* the wave function of the other particle (which I'll call B) will collapse and yield spin down. At first, I thought this was a problem: according to special relativity, simultaneity is relative, and so how can you talk about two events (the collapses of the wave functions of the two particles) being simultaneous in any, objective, meaningful way?

So what I did, was that I asked this exact question to professor Brian Greene in one of the live streams in his "Your Daily Equation"-series on YouTube. Here's a link to the video: <https://www.youtube.com/watch?v=iGDNzW59E3M> (at 39:18 in the video, he starts answering my question). Now, he said that the two collapses don't *have* to be simultaneous. It is just that we can choose a reference frame in which they are. But of course, if we instead go to another reference frame in uniform velocity compared to the other, then the collapses won't be simultaneous anymore, according to the rules of special relativity. But he did point out that since we can always choose *some* reference frame where the two events happen at the same time, the two events must be *spacelike* separated, and so no physical signal can ever reach particle B in time. Thus, the "spooky action at a distance"-business still applies.

But to me, there seems to be another problem too. For if we are in a frame where the two events are not simultaneous, then it will take some time for the wave function of particle B to collapse after the measurement of A, which we'll assume yielded spin up. So what if I then observe B *before* the collapse? Then, there is still a non-zero probability that B will yield spin up, and so that might actually happen. In other words, we might have that both particles yield spin up, which was supposed to be impossible. **What am I missing here?**