The minimal communication cost for simulating entangled qubits

We analyze the amount of classical communication required to reproduce the statistics of local projective measurements on a general pair of entangled qubits, |psi(AB)> = root p |00 > + root 1 - p |11 > (1/2 <= p <= 1). We construct a classical protocol that perfectly simulates local projective measurements on all entangled qubit pairs by communicating one classical trit. Additionally, when 2p(1-p)/2p-1 log (p/1-p) + 2(1 - p) <= 1, approximately 0.835 <= p <= 1, we present a classical protocol that requires only a single bit of communication. The latter model even allows a perfect classical simulation with an average communication cost that approaches zero in the limit where the degree of entanglement approaches zero ( p -> 1). This proves that the communication cost for simulating weakly entangled qubit pairs is strictly smaller than for the maximally entangled one.