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, $\mathrm{\mid }{\mathrm{\Psi }}_{AB}>=\sqrt{p}\mathrm{\mid }00>+\sqrt{1-p}\mathrm{\mid }11>$ (with $1\mathrm{/}2\le p\le 1$). We construct a classical protocol that perfectly simulates local projective measurements on all entangled qubit pairs by communicating one classical trit. Additionally, when $\frac{2p(1-p)}{2p-1}\log \left(\frac{p}{1-p}\right)+2(1-p)\le 1$, approximately $0.835\le p\le 1$, 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 ($p\to 1$). This proves that the communication cost for simulating weakly entangled qubit pairs is strictly smaller than for the maximally entangled one.