Analytical construction of (n, n-1) quantum random access codes saturating the conjectured bound

Quantum Random Access Codes (QRACs) embody the fundamental trade-off between the compressibility of information into limited quantum resources and the accessibility of that information, serving as a cornerstone of quantum communication and computation. In particular, the (n, n-1)-QRACs, which encode n bits of classical information into n-1 qubits, provides an ideal theoretical model for verifying quantum advantage in high-dimensional spaces; however, the analytical derivation of optimal codes for general n has remained an open problem. In this paper, we establish an analytical construction method for (n, n-1)-QRACs by using an explicit operator formalism. We prove that this construction strictly achieves the numerically conjectured upper bound of the average success probability, mathcal{P} = 1/2 + sqrt{(n-1)/n}/2, for all n. Furthermore, we present a systematic algorithm to decompose the derived optimal POVM into standard quantum gates. Since the resulting decoding circuit consists solely of interactions between adjacent qubits, it can be implemented with a circuit depth of O(n) even under linear connectivity constraints. Additionally, we analyze the high-dimensional limit and demonstrate that while the non-commutativity of measurements is suppressed, an information-theoretic gap of O\(log n\) from the Holevo bound inevitably arises for symmetric encoding. This study not only provides a scalable implementation method for high-dimensional quantum information processing but also offers new insights into the mathematical structure at the quantum-classical boundary.