When Are Popescu-Rohrlich Boxes and Random Access Codes Equivalent?

We study a problem of interconvertibility of two supraquantum resources: one is the so-called Popescu-Rohrlich (PR) box, which violates Clauser-Horne-Shimony-Holt inequality up to the maximal algebraic bound, and the second is the so-called random access code (RAC). The latter is a functionality that enables Bob (receiver) to choose one of two bits of Alice. It is known that a PR box supplemented with one bit of communication can be used to simulate a RAC. We ask the converse question: to what extent can a RAC can simulate a PR box? To this end, we introduce a "racbox": a box such that when it is supplemented with one bit of communication it offers a RAC. As said, a PR box can simulate a racbox. The question we raise is whether any racbox can simulate a PR box. We show that a nonsignaling racbox, indeed, can simulate a PR box; hence, these two resources are equivalent. We also provide an example of a signaling racbox that cannot simulate a PR box. We give a resource inequality between racboxes and PR boxes and show that it is saturated.