  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.
