AI 中文总结
研究二元(n,n - 1)和(n,n - 2)量子随机存取码的最优平均成功概率，利用林和德·沃尔夫的变换及相关约束，证明推测界，精确确定了这两种码的最优平均成功概率。
AI 中文摘要
二元(n,m)量子随机存取码将n位经典字符串压缩为m量子比特量子态，解码器从中恢复随机选择的目标比特。特别关注最优平均成功概率$P^{Q,\mathrm{avg},\mathrm{opt}}_{n,m}$，数值推测其满足$P^{Q,\mathrm{avg},\mathrm{opt}}_{n,m}\leq \frac{1}{2}+\frac{1}{2}\sqrt{\frac{m}{n}}$。近期铃木的(n,n - 1)码及秋部等人的(n,n - 2)码恰好达到此界，引发其是否严格最优的问题。本文通过证明m∈{n - 1,n - 2}时的推测上界解决了该问题，精确确定了$P^{Q,\mathrm{avg},\mathrm{opt}}_{n,n - 1}$和$P^{Q,\mathrm{avg},\mathrm{opt}}_{n,n - 2}$。证明利用了林和德·沃尔夫最近研究的通过相当好的测量从局部到全局重建的变换，以及对诱导信道的维度和半正定约束。
英文摘要
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, $P^{Q,\mathrm{avg},\mathrm{opt}}_{n,m}$, which is numerically conjectured to satisfy the bound $P^{Q,\mathrm{avg},\mathrm{opt}}_{n,m}\leq \frac{1}{2}+\frac{1}{2}\sqrt{\frac{m}{n}}$. 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 $P^{Q,\mathrm{avg},\mathrm{opt}}_{n,n-1}$ and $P^{Q,\mathrm{avg},\mathrm{opt}}_{n,n-2}$. 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.
