Robustness self-testing of states and measurements in the prepare-and-measure scenario with 3→1 random access code

Recently, Tavakoli et al. proposed a self-testing scheme in the prepare-and-measure scenario, showing that self-testing is not necessarily based on entanglement and violation of a Bell inequality[Phys. Rev. A 98 062307 (2018)]. They realized the self-testing of preparations and measurements in an N→1 (N ≥ 2) random access code (RAC), and provided robustness bounds in a 2→1 RAC. Since all N→1 RACs with shared randomness are combinations of 2→1 and 3→1 RACs, the 3→1 RAC is just as important as the 2→1 RAC. In this paper, we find a set of preparations and measurements in the 3→1 RAC, and use them to complete the robustness self-testing analysis in the prepare-and-measure scenario. The method is robust to small but inevitable experimental errors.