  Imagine that Alice and Bob, unable to communicate, are both given a 16-bit
string such that the strings are either equal, or they differ in exactly 8
positions. Both parties are then supposed to output a 4-bit string in such a
way that these short strings are equal if and only if the original longer
strings given to them were equal as well. It is known that this task can be
fulfilled without failure and without communication if Alice and Bob share 4
maximally entangled quantum bits. We show that, on the other hand, they CANNOT
win the same game with certainty if they only share classical bits, even if it
is an unlimited number. This means that for fulfilling this particular
distributed task, quantum entanglement can completely replace communication.
This phenomenon has been called pseudo-telepathy. The results of this paper
complete the analysis of the first proposed game of this type between two
players.
