What happens if we measure the two entangled systems of the Bell test simultaneously?

Question

Consider the Bell state $|\psi\rangle =\frac{|00\rangle +|11\rangle}{\sqrt{2}}$.

When measuring one of the individual systems, the superposition is projected onto one of the eigenstates $|00\rangle$ or $|11\rangle$.

Imagine we measured the entangled systems at the same time. Which part will project the state? That is, at measurement one part of the system determines the outcome of the other. But if measured at the same time, which determines the outcome?

For the moment I neglect special relativity and think the observer to be in the rest frame of the experiment.

Answer 1

The outcome is determined (probabilistically) by the state, and is either $|00\rangle$ or $11\rangle$ (assuming that's the observation you're making) equiprobably.

Answer 2

To add on to WillO's answer, I think the key in entanglement is not "which measurement happens first" or "which measurement affects the other", but rather that $|00\rangle$ and $11\rangle$ are the only possible outcomes! No matter what frame you switch to, you will NEVER get $|01\rangle$ or $|10\rangle$.

Even if you consider the different frames in which the two events can have different chronological order, the Bell test still rules out local hidden variable theories because the two events are causally disconnected regardless of which frame you look at it from. So the causal disconnection is not affected by the chronological ambiguity.