Improved classical and quantum random access codes

Abstract

A (quantum) random access code ((Q)RAC) is a scheme that encodes $n$ bits into $m$ (qu)bits such that any of the $n$ bits can be recovered with a worst case probability $p>\frac{1}{2}$. We generalize (Q)RACs to a scheme encoding $n$ $d$-levels into $m$ (quantum) $d$-levels such that any $d$-level can be recovered with the probability for every wrong outcome value being less than $\frac{1}{d}$. We construct explicit solutions for all $n\ensuremath{\le}\frac{{d}^{2m}\ensuremath{-}1}{d\ensuremath{-}1}$. For $d=2$, the constructions coincide with those previously known. We show that the (Q)RACs are $d$-parity oblivious, generalizing ordinary parity obliviousness. We further investigate optimization of the success probabilities. For $d=2$, we use the measure operators of the previously best-known solutions, but improve the encoding states to give a higher success probability. We conjecture that for maximal $(n={4}^{m}\ensuremath{-}1,m,p)$ QRACs, $p=\frac{1}{2}{1+{[{(\sqrt{3}+1)}^{m}\ensuremath{-}1]}^{\ensuremath{-}1}}$ is possible, and show that it is an upper bound for the measure operators that we use. We then compare $(n,m,{p}_{q})$ QRACs with classical $(n,2m,{p}_{c})$ RACs. We can always find ${p}_{q}\ensuremath{\ge}{p}_{c}$, but the classical code gives information about every input bit simultaneously, while the QRAC only gives information about a subset. For several different $(n,2,p)$ QRACs, we see the same trade-off, as the best $p$ values are obtained when the number of bits that can be obtained simultaneously is as small as possible. The trade-off is connected to parity obliviousness, since high certainty information about several bits can be used to calculate probabilities for parities of subsets.