Optimal Average Success Probabilities of Binary $(n,n-1)$ and $(n,n-2)$ Quantum Random Access Codes via a Proof of the Corresponding Conjectured Bound

A binary $(n,m)$ quantum random access code (QRAC) compresses an $n$-bit classical string into an $m$-qubit quantum state, from which a decoder attempts to recover a randomly selected target bit. Of particular interest is the optimal average probability of success, $P_{n,m}^{Q,\mathrm{a}\mathrm{v}\mathrm{g},\mathrm{o}\mathrm{p}\mathrm{t}}$, which is numerically conjectured to satisfy the bound $P_{n,m}^{Q,\mathrm{a}\mathrm{v}\mathrm{g},\mathrm{o}\mathrm{p}\mathrm{t}}\le \frac{1}{2}+\frac{1}{2}\sqrt{\frac{m}{n}}$. Recent constructions of $(n,n-1)$ QRACs by Suzuki and $(n,n-2)$ QRACs by Akibue et al. meet this bound exactly, raising the question of their strict optimality. In this work, we settle this question by proving the conjectured upper bound for $m\in \{n-1,n-2\}$, thereby precisely determining $P_{n,n-1}^{Q,\mathrm{a}\mathrm{v}\mathrm{g},\mathrm{o}\mathrm{p}\mathrm{t}}$ and $P_{n,n-2}^{Q,\mathrm{a}\mathrm{v}\mathrm{g},\mathrm{o}\mathrm{p}\mathrm{t}}$. The proof utilizes a translation recently studied by Lin and de Wolf from local to global reconstruction via pretty good measurement, along with dimensional and positive-semidefinite constraints on an induced channel.