Sufficient conditions for quantum advantage in random access code protocols with two-qubit states

Random access code (RAC) is an important communication protocol to obtain information about a randomly specified substring of an n-bit string, while only having limited information about the n-bit string. Quantum RACs usually utilise either communication of quantum bits or a shared-in-advance quantum state used in conjunction with classical communication. Here we consider the latter version of the quantum protocols under the constraint of single-bit communication and with shared arbitrary state of two qubits. Taking the worst-case success probability as the figure of merit, we demonstrate that any state with invertible correlation matrix can be used to outperform the best classical RAC for n=3. We derive an additional condition sufficient to beat the best classical performance in the case of n=2. In particular, separable states turn out to be a useful resource behind the quantum advantage for n=2,3. For $n \geq 4$ RACs assisted with a single copy of a quantum state do not outperform the classical RACs.