Random access codes and non-local resources

It is known that a PR-BOX (PR), a non-local resource and \(2→ 1\) 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 \(2→ 1\) RAC and study their inter-convertibility. We introduce generalizations based on the number of inputs provided to Alice, B_n-BOX and \(n→ 1\) RAC. We show that a B_n-BOX is equivalent to a no-signaling \(n→ 1\) RACBOX (RB). Further we introduce a signaling \(n→ 1\) RB which cannot simulate a B_n-BOX. Finally to quantify the same we provide a resource inequality between \(n→ 1\) 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 \(n→ 1\) 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 \(n→ 1,d\) 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 \(n→ 1,d\) 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 \(n→ 1,d\) RB and B_n^d(+)-BOX, and show that it is saturated.