Mutually unbiased balanced functions and generalized random access codes

Abstract

Quantum resources and protocols are known to outperform their classical counterparts in variety of communication and information processing tasks. Random Access Codes (RACs) are one such cryptographically significant family of bipartite communication tasks, wherein, the sender encodes a data set (typically a string of input bits) onto a physical system of bounded dimension and transmits it to the receiver, who then attempts to guess a randomly chosen part of the sender's data set (typically one of the sender's input bits). In this work, we introduce a generalization of this task wherein the receiver, in addition to the individual input bits, aims to retrieve randomly chosen functions of sender's input string. Specifically, we employ sets of mutually unbiased balanced functions (MUBS), such that perfect knowledge of any one of the constituent functions yields no knowledge about the others. We investigate and bound the performance of (i.) classical, (ii.) quantum prepare and measure, and (iii.) entanglement assisted classical communication (EACC) protocols for the resultant generalized RACs (GRACs). Finally, we detail the case of GRACs with three input bits, find maximal success probabilities for classical, quantum and EACC protocols, along with the effect of noisy quantum channels on the performance of quantum protocols. Moreover, with the help of this case study, we reveal several characteristic properties of the GRACs which deviate from the standard RACs.