If the up-down relationship is proven at a distance with entanglement, why isn’t just one of the two bits sufficient to give the information needed? In other words if the state of A is up, then you know the state of B. It is faster than light communication, if you know the original state. Picture carrying A a light year away, with the information regarding B’s original state. Wouldn’t changes occur faster than light?
Nope, I’ll try again with a little more math this time.
First, the intermediary entangles two quantum states and sends them to A and B. Now, A has a 50% chance of measuring 1 and 50% chance of measuring 0. Same for B. However, whatever measurement A gets, B must get the same thing. In the frame of either of the two scientists, they’re still getting 50/50 output and can’t determine anything about the next one (unless told by the other scientist). But when they get together and compare notes, they find their data agrees perfectly. (It’s like they’re both flipping the same coin...)
When A performs her operation on her bit, she changes the phase of the entanglement (note that information is contained in the phase again - another signature of quantum mechanics). She can transform the pair to one of four orthogonal states of entanglement. They both still have a 50% chance of measuring 1 and 50% chance of measuring 0. Depending on the operation she made, she can make Bob’s bit match hers or not. However, since her measurement is still 50/50, B can’t tell what she did by measuring his bit (without knowing what her bit was).
If A measures hers and calls B to tell him about it, they can either match or not - so he gets one bit of information (this is pointless, it’s equivalent to the classical case). On the other hand, if she sends him her bit (before she measures it), then he posses the entire entangled state which can be in one of four states. By measuring the state of the entanglement (via the Bell basis), he can get 1 of 4 outcomes - or two bits of information - and that is superdense coding.
Note that above there is no superluminal communication. Also note the careful use of measurement: If either one of them measures their bit, the entanglement collapses and we’re back to classical. This is another signature of quantum mechanics - measurement affects the outcome. Bob either needs to know Alice’s bit (classical communication), or Alice needs to send him her bit (superdense coding).
Without a background in QM, it’s pretty difficult to grasp.