Imagine Alice and Bob have two pairs of entangled particles, pair A, and pair B. Imagine they have agreed beforehand that pair A measurement preceding pair B measurement constitutes a bit with value 1 and pair B measurement preceding pair A measurement represents a 0 value bit. Would they not be able to use this predetermined order to transmit at least one bit information instantaneously across large distances?
Where is the flaw in this chain of reasoning?
The quantum mechanical expectation value (prediction) is generally independent of order when there is entanglement. Whether Alice measured first or not has no observable difference to the outcomes as far as anyone knows. The outcomes Alice alone sees are always random. Same with Bob. Not much information to be gained from looking at a random stream of bits.
Suppose Alice measures first. She measures A then B to send a 1.
What does Bob do? He measures both in either order. He gets answers. How does he know the order Alice made her measurements? Or if she has meade measurements?
If A before B, then one of them, say Alice, gets 1 bit. Bob doesn't know any thing. He could measure in any order. Only if Alice sends a message to Bob telling him that she got 1 bit, Bob will then know what to do and will be able to predict what he's going to get. But notice, this was only due to a conventional communication that took place no faster than the speed of light.
Alice and Bob are living in a relativistic universe. Who measures first is observer dependent. This is usually called the Andromeda paradox. Since all observers have to agree with each other about the outcome of those measurements, the order of events can not convey any kind of information, even if it can be correlated. The main problem with the discussion of quantum mechanics at the level of entanglement is that it drags the (incorrect) structure of Galilean spacetime into the picture by analyzing it exclusively in the center of mass system of the two measurements.
I am not a theorist, but intuitively it seems to me that quantum mechanics is a relativistic phenomenon. There can be no self-consistent non-relativistic version of it and if we look at the endless and fruitless discussions of these phenomena at the level of non-relativistic theory nature seems to back that notion up.
