Certified quantum supremacy in entanglement-assisted prepare-measure random-access-code

We develop a family of semi-device-independent (SDI) entanglement-assisted prepare-measure (PM) communication games involving two parties, within the $n\rightarrow l$ random-access code (RAC) framework where the sender Alice holds a n-bit string and communicates $l<n$ bits or qubits to the receiver Bob. In contrast to the standard quantum PMRAC, here the parties share a prior entanglement, and Alice applies quantum operations on her sub-system to encode her inputs and sends to Bob. We first consider the $4\rightarrow l$ entanglement-assisted PMRAC with $l=1$ and $2$ and derive the optimal quantum success probabilities using an elegant analytical technique. We demonstrate quantum supremacy over both classical RACs and conventional quantum PMRACs. Moreover, we exhibit that the optimal quantum advantage allows one to certify Alice's unitary operations. We then derive an upper bound on the quantum success probabilities for $5\rightarrow l$ entanglement-assisted PMRAC with $l=1,2$ and $3$. Further, we extend the demonstration of quantum advantage for $n\rightarrow n-2$ case where n is arbitrary.