A separable version of Ky Fan's majorization relation is proven for a sum of two operators that are each a tensor product of two positive semi-definite operators. In order to prove it, upper bounds are established for the relevant largest eigenvalue sums in terms of the optimal values of certain linear programs. The objective function of these linear programs is the dual of the direct sum of the spectra of the summands. The feasible sets are bounded polyhedra determined by positive numbers, called alignment terms, that quantify the overlaps between pairs of largest eigenvalue spaces of the summands. By appealing to geometric considerations, tight upper bounds are established on the alignment terms of tensor products of positive semi-definite operators. As an application, the spin alignment conjecture in quantum information theory is affirmatively resolved to the 2-letter level. Consequently, the coherent information of platypus channels is additive to the 2-letter level.

翻译：针对两个算子的和，证明了Ky Fan主控关系的可分版本，其中每个算子均为两个半正定算子的张量积。为证明该结论，通过特定线性规划最优值的形式，建立了相关最大特征值之和的上界。这些线性规划的目标函数是求和项谱直和对偶。可行集是由正数界定的有界多面体，这些正数称为对齐项，用于量化求和项最大特征值空间对之间的重叠程度。通过几何考量，建立了半正定算子张量积对齐项的紧上界。作为应用，量子信息论中的自旋对齐猜想在2字母层级得到肯定性解决。因此，鸭嘴兽信道的相干信息在2字母层级具有可加性。