Quantum Random Access Codes with Shared Randomness

TL;DR

This paper explores quantum random access codes enhanced with shared randomness, establishing bounds on success probabilities, comparing quantum and classical models, and providing constructions for small cases.

Contribution

It introduces shared randomness into quantum random access codes, derives success probability bounds, and compares quantum and classical encoding strategies.

Findings

Shared randomness allows (n,1,p) QRACs for any n with p > 1/2.

Upper bounds on success probabilities match known QRACs for small n.

Classical codes with shared randomness have lower success probabilities than quantum codes.

Abstract

We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not possible. We extend this model with shared randomness (SR) that is accessible to both parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79) QRACs match this upper bound). We discuss some particular constructions for several small values of n. We also study the classical counterpart of this model where n bits are encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal construction for such codes and find their success probability exactly--it is less than in the quantum case. Interactive 3D quantum random access codes are available on-line at http://home.lanet.lv/~sd20008/racs .