The famous EPR (Einstein-Podolsky-Rosen) paradox describes a rather counterintuitive quality of quantum physics: the quantum state space of a set of several particles is the direct product of their individual spaces, and not the direct sum as one would expect. As a consequence, if A and B have separate particles a and b, which happen to be in an entangled state, then A may be able to gain information on B's measurement of b by measuring a.
It is possible for A to exploit this in order to cheat. Here's A's cheating scheme:
When Alice measures her qubit with respect to the horizontal basis, she gets
either a zero or a one with 50% chance. However, this projects the entire state
space of the entangled pair to either
components are zero from
the start), which means that Bob will obtain the same result5. This is exactly the EPR paradox (table 4).
|qubit pair||A's basis||new state||B's basis||P(0)||P(1)|
With this procedure Alice makes sure that whenever Bob used the same basis to do his measurement, so he will get an identical result. And in the opposite case, as the angle between state and basis is , he obtains a 50%-50% distribution as expected.