Classical codes violate the conjectured square-root bound for quantum random access codes

We consider whether every quantum random access code (QRAC) with density-operator encodings and arbitrary decoding measurements obeys the conjectured bound $p\leq(1+\sqrt{m/n})/2$, where $n$ classical bits are encoded into $m$ qubits and $p$ is the worst-case success probability.

We find that classical random access codes with private randomness, which form a subclass of this QRAC model, violate the bound.

We embed these classical codes as QRACs with diagonal encoding states and commuting decoding measurements, and construct pure-state realizations with identical decoding statistics.

The achievability theorem of Ambainis, Nayak, Ta-Shma, and Vazirani then yields violations for every fixed $p\in(1/2,1)$ at sufficiently large input length.

The counterexamples span the full open interval between the conjectured and Nayak bounds at each fixed compression rate.

A finite-blocklength analysis further yields order-optimal logarithmic qubit scaling for a recovery bias scaling as $\sqrt{\log_2 n/n}$ with a sufficiently large prefactor.

These results identify the classical coding rate as the source of the separation and motivate restricted bounds based on quantitative spectral properties of decoding measurements.