Mutually Unbiased Balanced Functions & Generalized Random Access Codes

TL;DR

This paper introduces a generalized form of Random Access Codes using mutually unbiased balanced functions, analyzing classical, quantum, and entanglement-assisted protocols, and explores their performance and properties.

Contribution

It presents a novel generalization of RACs with mutually unbiased functions and analyzes their performance across different communication protocols.

Findings

Quantum protocols outperform classical ones in GRACs.

Maximal success probabilities are derived for three-input-bit GRACs.

Noisy channels impact quantum protocol performance.

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.