互补量子关联的可分反例，以及随机搜索为何未能发现它们
Separable Counterexamples to Complementary Quantum Correlations, and Why Random Search Missed Them
AI总结：
该研究反驳了互补量子关联（CQC）关系，构造了多维度对下的可分反例，分析了剩余量子比特-量子维维度的特性，给出普适修正不等式，并解释了随机搜索未能发现反例的原因。
AI中文摘要：
互补量子关联（CQC）关系通过测量前态的量子互信息，对通过局部相互无偏测量得到的两个经典互信息之和进行了界定。我们对该关系进行了反驳：在每一对满足m,n≥3的局部维度对m×n中，都存在秩为2的可分反例，其闭式超出量至少为1/(8m²n²)纳特；在每一对量子比特-量子维对2×n（n≥3，除n=3,5外）中也存在此类反例，且所有覆盖的维度对均存在满秩可分反例。随后我们分析了剩余的两对量子比特-量子维维度：一种无维度熵包络替代了自然二次优控项，将闭合等先验正交两射线族的要求从三角判别界S≤8/3降至S≤3.8265583…；相关门矩阵的迹恰好为2，因此其谱检验坍缩为无特征值的标量；精确恒等式X=1−4Var(c)将素傅里叶全火花壁垒转化为方差界。128位区间覆盖在2×3处闭合该族，间隙至少为0.012021纳特；在2×5处，达到S=(14+2√5)/5的精确饱和器反驳了三条竞争路径。我们还给出了一种依赖态的普适修正不等式，并表明其在2×2处已与CQC不可比。最后我们量化了原始搜索为何基本无力发现这些态：违反集是低秩边界上的细条，见证算子比希尔伯特-施密特均值低7.3个标准差，基必须对齐至约9度（约为框架的10⁻²¹），而外推样本最小值需要10¹⁰至10¹⁸个样本，远超过已运行的10⁷个样本。
英文摘要：
The complementary quantum correlations (CQC) relation bounds the sum of two classical mutual informations, obtained from local mutually unbiased measurements, by the quantum mutual information of the premeasurement state. We refute it. Separable rank-two counterexamples exist in every local dimension pair \(m\times n\) with \(m,n\ge3\), with closed-form excess at least \(1/(8m^2n^2)\) nats, and in every qubit--qudit pair \(2\times n\) with \(n\ge3\) except \(n=3,5\); every covered pair also admits full-rank separable counterexamples. We then analyse the two residual qubit--qudit dimensions. A dimension-free entropy envelope replaces the natural quadratic majorant and lowers the requirement for closing the equal-prior orthogonal two-ray family from a triangular-discrimination bound \(S\le8/3\) to \(S\le3.8265583\ldots\); the associated gate matrix has trace exactly two, so its spectral test collapses to a single eigenvalue-free scalar; and the exact identity \(X=1-4\operatorname{Var}(c)\) turns the prime-Fourier full-spark barrier into a variance bound. A \(128\)-bit interval cover then closes that family at \(2\times3\) with gap at least \(0.012021\) nats, and at \(2\times5\) an exact saturator attaining \(S=(14+2\sqrt5)/5\) refutes three competing routes. We also give a state-dependent corrected inequality that is universal, and show it is incomparable with CQC already at \(2\times2\). Finally we quantify why the original searches had essentially no power to find these states: the violating set is a sliver against the low-rank boundary, the witness lies \(7.3\) standard deviations below the Hilbert--Schmidt mean, the bases must be aligned to about nine degrees (\(\sim10^{-21}\) of frames), and extrapolating the sample minimum demands \(10^{10}\) to \(10^{18}\) samples against the \(10^7\) ever run.
