发表机构
Tohoku University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发展量子结合方案的Delsarte理论，给出码与设计的锥界，并证明量子Johnson方案中强度二设计的最小有效大小为20。
AI 中文摘要
我们为量子结合方案发展了Delsarte理论，允许复合和量子Schur积均为非交换。MacWilliams恒等式以及内分布和偶分布的 positivity 给出了码的锥上界和设计的锥下界，这些界以有效大小（有效大小扩展了子集基数）表示。这些界恢复了对称结合方案的经典线性规划和齐次相干构型的半正定规划。我们还利用有限群表示量子化Schur方案，并通过在逐个固定指定点的子群上取平均，刻画了由不可约稳定子表示产生的量子Johnson方案中的量子块设计。在$M_2(\mathbb{C})^{\oplus10}$上的一个量子Johnson方案中，我们证明了满足强度二设计条件的非零正元素的最小有效大小为$20$，该下界由基础代数中心之外的互补投影达到。中心中的每个此类元素的有效大小为$40$，而锥下界为$40/3$，严格低于实际最小值。
英文摘要
We develop Delsarte theory for quantum association schemes, allowing both composition and the quantum Schur product to be noncommutative. A MacWilliams identity and positivity of the inner and dual distributions yield conic upper bounds for codes and lower bounds for designs in terms of effective size, which extends subset cardinality. These bounds recover classical linear programs for symmetric association schemes and semidefinite programs for homogeneous coherent configurations. We also quantize Schurian schemes using finite-group representations and characterize quantum block designs in quantum Johnson schemes arising from irreducible stabilizer representations by averaging over subgroups fixing prescribed points individually. In a quantum Johnson scheme on $M_2(\mathbb{C})^{\oplus10}$, we prove that the minimum effective size of a nonzero positive element satisfying the strength-two design condition is $20$, attained by complementary projections outside the center of the underlying algebra. Every such element in the center has effective size $40$, while the conic lower bound is $40/3$, strictly below the actual minimum.
Comments46 pages, 1 figure
