Random access codes and nonlocal resources

Abstract

It is known that a PR-BOX (PR), a non-local resource and $(2\rightarrow 1)$ random access code (RAC), a functionality (wherein Alice encodes 2 bits into 1 bit message and Bob learns one of randomly chosen Alice's inputs) are equivalent under the no-signaling condition. In this work we introduce generalizations to PR and $(2\rightarrow 1)$ RAC and study their inter-convertibility. We introduce generalizations based on the number of inputs provided to Alice, $B_n$-BOX and $(n\rightarrow 1)$ RAC. We show that a $B_n$-BOX is equivalent to a no-signaling $(n\rightarrow 1)$ RACBOX (RB). Further we introduce a signaling $(n\rightarrow 1)$ RB which cannot simulate a $B_n$-BOX. Finally to quantify the same we provide a resource inequality between $(n\rightarrow 1)$ RB and $B_n$-BOX, and show that it is saturated. As an application we prove that one requires atleast $(n-1)$ PRs supplemented with a bit of communication to win a $(n\rightarrow 1)$ RAC. We further introduce generalizations based on the dimension of inputs provided to Alice and the message she sends, $B_n^d(+)$-BOX, $B_n^d(-)$-BOX and $(n\rightarrow 1,d)$ RAC ($d>2$). We show that no-signaling condition is not enough to enforce strict equivalence in the case of $d>2$. We introduce classes of no-signaling $(n\rightarrow 1,d)$ RB, one which can simulate $B_n^d(+)$-BOX, second which can simulate $B_n^d(-)$-BOX and third which cannot simulate either. Finally to quantify the same we provide a resource inequality between $(n\rightarrow 1,d)$ RB and $B_n^d(+)$-BOX, and show that it is saturated.