Quantum Advantages in (n,d)->1 Random Access Codes

A random access code (RAC), corresponding to a communication primitive with various applications in quantum information theory, is an instance of a preparation-and-measurement scenario. In this work, we consider (n,d)-RACs constituting an "n"-length string, constructed from a "d" size set of letters, and send an encoding of the string in a single d-level physical system and present their quantum advantages. We first characterize optimal classical RACs, proving that the well-known classical strategy known as majority-encoding-identity-decoding is indeed optimal. We then construct a quantum protocol by exploiting only two incompatible measurements, the minimal requirement, and show the advantages beyond the classical one. We also discuss the generality of our results and whether quantum advantages are valid for all types of (n, d)->1 RACs.