2026/07/17 by Kangqiao Liu
#quant-ph #cs.IT #math.IT
We consider whether every quantum random access code (QRAC) with density-operator encodings and arbitrary decoding measurements obeys the conjectured bound p≤(1+√(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∈(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 √(log2 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.