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, |Ψ_AB〉 = √p |00〉 + ^√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 ^2p₂^{(1 _p−−₁ p)} log ^(₁₋p_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.