Another Bell's Theorem Question
I am trying to follow the simple model of Bell's Theorem outlined in this paper: https://people.eecs.berkeley.edu/~vazirani/s07quantum/notes/lecture1.pdf. Please read section 5 for details, but basically he outlines a communication protocol where two people $A$ and $B$ receive a bit each ($X_a$ and $X_b$). They than each have to independently produce new bits ($a$ and $b$). $A$ and $B$ are trying to cooperatively maximize the probability that $$a\ XOR\ b = X_a\ AND\ X_b$$
There is a trivial strategy that wins 75 percent of the time. Always produce $a$ and $b$ of 0. The last section describes the strategy that wins more than 75% of the time in quantum mechanics and disproves 'local hidden variables'. From my understanding a local hidden variable theory would be one where the two particles before they were separated planned out every possible result of any possible experiment that could be performed once they are separated. This way there was no FTL communication.
My questions are as follows:
If in the quantum version of the game they always have the EPR pair, than why can't they win 100 percent of the time? If A measures 1 he produces 1 and B always produces 0. I don't get what this has to do with quantum mechanics since this strategy could be devised even if hidden variable theory was 'true' and the two particles 'agreed' on there configurations before they were separated in space.
In the protocol at the end he says if $X_a$ = 0 than do a certain measurement. What is the output produced by the strategy? It is very unclear. Is the output the result of the measurement?
Also don't you have to perform a measurement to know that $X_a$ = 0? Wouldn't this already affect the state EPR pair before you made the second measurement?
Am I fundamentally missing something?