Random access codes and nonlocal resources

Abstract

This work explores the notion of inter-convertibility between a cryptographic primitive: the random access code (RAC) and bipartite no-signaling nonlocal resources. To this end we introduce two generalizations of the Popescu-Rohrlich box (PR) and investigate their relation with the corresponding RACs. The first generalization is based on the number of Alice's input bits; we refer to it as the ${B}_{n}$-box. We show that the no-signaling condition imposes an equivalence between the ${B}_{n}$-box and the $(n\ensuremath{\rightarrow}1)$ RAC (encoding of $n$ input bits to 1 bit of message). As an application we show that $(n\ensuremath{-}1)$ PRs supplemented with one bit communication are necessary and sufficient to win a $(n\ensuremath{\rightarrow}1)$ RAC with certainty. Furthermore, we present a signaling instant of a perfectly working $(n\ensuremath{\rightarrow}1)$ RAC which cannot simulate the ${B}_{n}$-box, thus showing that it is weaker than its no-signaling counterpart. For the second generalization we replace Alice's input bits with $d\mathrm{its}$ ($d$-leveled classical systems); we call this the ${B}_{n}^{d}$-box. In this case the no-signaling condition is not enough to enforce an equivalence between the ${B}_{n}^{d}$-box and $(n\ensuremath{\rightarrow}1,d)$ RAC (encoding of $n$ input $d\mathrm{its}$ to $1\phantom{\rule{4pt}{0ex}}d\mathrm{it}$ of message); i.e., while the ${B}_{n}^{d}$-box can win a $(n\ensuremath{\rightarrow}1,d)$ RAC with certainty, not all no-signaling instances of a $(n\ensuremath{\rightarrow}1,d)$ RAC can simulate the ${B}_{n}^{d}$-box. We use resource inequalities to quantitatively capture these results.