Some physicists currently understand entanglement as transferring information instantaneously, yet not violating causality. Is this really a satisfactory explanation, or should be look for something better?
In particular, some people try to be explicit about a signalling between particles, but that does not seem consistent with relativity.
For example, if you have two particles, entangled and spatially separated, then the results of experiments can be predicted easily by a local hidden variable theory if the experimenters are forced to pick known (and equal) choices about what to measure (out of complementary variables, such as the $x$-component of spin and the $y$-component of spin).
So let's consider the situation where there are two labs (Alice's Lab to measure particle A, Bob's lab to measure particle B), and they are selecting (potentially) different things to measure. You might postulate that if particle B were measured first, the result $a$ produced by particle A needs to be a function $a = a(X, Y, b)$, where $X$ is the type of measurement done on A, $Y$ the type done on B, and $b$ the result produced by B at the measurement $Y$. Otherwise it is hard to agree with the correlations required.
Similarly, if particle A were measured first, then the result $b = b(X, Y, a)$. Otherwise it is hard to agree with the correlations required. But if you need the results of the other lab to generate consistent results here, how can the data from about $Y$ and $b$ be available to Alice's Lab?
Some people propose (and I argue against) that particles can transmit their results Faster than Light (FTL) to the other particle. In order to get the right results, it looks like the alleged faster than light protocol would go like:
We have two entangled particles A and B. A flies to a lab where the experimenter Alice works, while B flies to the lab of experimenter Bob.
The protocol is as follows:
Alice measures her particle, A.
Particle A transmits to particle B, by superluminal signals, the following information:
a) which measurement was done on A;
b) which result the particle A produced.
The experimenter Bob measures his particle, B.
Particle B transmits to particle A, by superluminal signals, the following information:
c) which measurement was done on B.
d) which result the particle B produced.
I don't get into details of which emitter/receiver A and B possess, and how many bits it takes to describe a type of experiment. I will just say that on the Earth the two experimenters appear to be done, at each trial, simultaneously. By standard relativity arguments, if these events are spacelike separated different observers can disagree about which happened first, so there isn't a consistent story about which one happened first, and which sent information to the other. The whole superluminal signalling program seems to be not enough to explain entanglement. To provide the details, consider the following.
I'll focus on two travellers, Charlie and Dan. Charlie travels in a rocket in the direction of Alice, and Dan in a rocket in the direction of Bob. To an observer on Earth, their velocity is equal in absolute value. Therefore, by Charlie's clock, Alice measures in each trial before Bob, while by Dan's clock, Bob measures in each trial before Alice.
Now, let's ask Charlie what he can say about the above protocol: Particle A indeed sends to particle B, all the signals with info about the type of measurement done and the result. But, at the same time it sends to B all the signals with which type of experiment is done on B, and with which result. That, because superluminal signals that appear in one frame as sent from B to A, appear in another frame as sent from A to B.
But, no matter from whom to whom they are sent, the price to be paid if one accept this protocol is that at Alice's site and time-of-measurement, it's known which type of measurement Bob will choose, even before Bob will at all make a decision about which type of measurement to choose.
Thus, what should we do? Assume that a particle is endowed with prophecy about people's decision? Or, simply say that we don't yet understand properly how entanglements work, and as Danu formulated it better, to seek another explanation ?
Note: I would appreciate not to suggest changing the premises of the problem, e.g. not to suggest that the apparatus in one labs measures the particle in the other lab, or to disconsider relativity. And also not to propose me fuzzy ideas of the type (this or that) is done somehow, neither to suggest me what would be if some laws of the physics could be overridden. I would also appreciate to read the text before commenting it, and also to read the discussion that was done until now in the comments.