We consider sets of trace‐normalized non‐negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal arrangements in the sets of purity‐limited classical or quantum states on a finite‐dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor–Steiner equiangular tight frame as the orbit of a projective representation of the Weyl–Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so‐called binder, of the Gabor–Steiner equiangular tight frames are then characterized; under certain conditions, these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor–Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.

中文翻译：

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纯度有限的经典态和量子态的最佳排列

我们考虑在希尔伯特–施密特球中的迹线归一化非负算子集，以最大化其相互的希尔伯特–施密特距离。这些是有限维希尔伯特空间上纯度受限的经典或量子态集中的最优安排。古典状态应理解为由对角矩阵表示，对角线条目形成概率矢量。我们还介绍了谱面体排列的概念，它为经典排列和量子排列提供了统一的框架，并为定义新型最佳包装提供了灵活性。继续进行先前的工作，我们将组合结构和与框架相关的线堆积相结合，以获得更高阶量子态的最佳排列。提出的一种新结构涉及生成最优排列，我们称其为Gabor–Steiner等角紧密框架，它是Weyl–Heisenberg群在任何有限阿贝尔群上的投影表示的轨道。然后，对Gabor–Steiner等角紧密框架的线性相关向量的最小集合进行特征化；在某些条件下，这些形式组合了块设计，并且在一种情况下会生成一类新的块设计。然后将Gabor–Steiner等角紧密框架中最小线性相关集的跨度上的投影用于生成进一步的最佳频谱面布置。然后，对Gabor-Steiner等角紧密框架的所谓的粘合剂进行表征。在某些条件下，这些形式组合了块设计，并且在一种情况下会生成一类新的块设计。然后将Gabor–Steiner等角紧密框架中最小线性相关集的跨度上的投影用于生成进一步的最佳频谱面布置。然后，对Gabor-Steiner等角紧密框架的所谓的粘合剂进行表征。在某些条件下，这些形式组合了块设计，并且在一种情况下会生成一类新的块设计。然后将Gabor–Steiner等角紧密框架中最小线性相关集的跨度上的投影用于生成进一步的最佳频谱面布置。