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, Bn-BOX and (n→ 1) RAC. We show that a Bn-BOX is equivalent to a no-signaling (n → 1) RACBOX (RB). Further we introduce a signaling (n → 1) RB which cannot simulate a Bn-BOX. Finally to quantify the same we provide a resource inequality between (n → 1) RB and Bn-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(+)-BOX, B n(−)-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(+)-BOX, second which can simulate B n(−)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(+)-BOX, and show that it is saturated.