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

2026-07-17

AI Summary

This work investigates whether quantum random access codes satisfy the conjectured square-root bound $p \leq (1 + \sqrt{m/n})/2$. By embedding classical random access codes with private randomness into quantum schemes using diagonal encoding states and commuting POVM measurements, the authors construct the first classical counterexamples that violate this bound, revealing that classical coding rates are key to the separation from the quantum limit. Leveraging spectral properties of decoding measurements, they establish a new restricted bound and prove that for any fixed $p \in (1/2, 1)$, counterexamples exist when the input length is sufficiently large. Moreover, they achieve optimal logarithmic qubit scaling when the recovery bias is $\sqrt{\log_2 n / n}$.

Abstract

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.

Problem

• quantum random access codes
• square-root bound
• classical codes
• decoding measurements
• success probability

Innovation

• quantum random access codes
• square-root bound violation
• classical coding rate
• diagonal encoding states
• commuting measurements