Quantum Random Access Codes for Boolean Functions

An n?pm random access code (RAC) is an encoding of xmlns:mml="http://www.w3.org/1998/Math/MathML">n bits into xmlns:mml="http://www.w3.org/1998/Math/MathML">m such that any initial bit can be recovered with probability at least xmlns:mml="http://www.w3.org/1998/Math/MathML">p, while in a quantum RAC (QRAC), the are encoded qubits. Since its proposal, idea RACs was generalized many different ways, e.g. allowing use shared entanglement (called entanglement-assisted code, or simply EARAC) recovering multiple instead one. In this paper we generalize to value given Boolean function xmlns:mml="http://www.w3.org/1998/Math/MathML">f on subset fixed size bits, which call xmlns:mml="http://www.w3.org/1998/Math/MathML">f-random codes. We study and give protocols for codes classical (f-RAC) xmlns:mml="http://www.w3.org/1998/Math/MathML">f-QRAC) encoding, together resources, private randomness, xmlns:mml="http://www.w3.org/1998/Math/MathML">f-EARAC) Popescu-Rohrlich boxes xmlns:mml="http://www.w3.org/1998/Math/MathML">f-PRRAC). The success our characterized by xmlns:mml="http://www.w3.org/1998/Math/MathML">noise stability xmlns:mml="http://www.w3.org/1998/Math/MathML">f. Moreover, mathvariant="italic">upper bound xmlns:mml="http://www.w3.org/1998/Math/MathML">f-QRAC randomness matches up multiplicative constant (and xmlns:mml="http://www.w3.org/1998/Math/MathML">f-RACs extension), meaning only achieve limited advantage over their counterparts.