  We consider general prepare-and-measure scenarios in which Alice can transmit
qubit states to Bob, who can perform general measurements in the form of
positive operator-valued measures (POVMs). We show that the statistics obtained
in any such quantum protocol can be simulated by the purely classical means of
shared randomness and two bits of communication. Furthermore, we prove that two
bits of communication is the minimal cost of a perfect classical simulation. In
addition, we apply our methods to Bell scenarios, which extends the well-known
Toner and Bacon protocol. In particular, two bits of communication are enough
to simulate all quantum correlations associated to arbitrary local POVMs
applied to any entangled two-qubit state.
