We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for H^d determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of R^d(2d+1) of dimension 3.

翻译：我们研究了三种不同环境的紧凑投影 2 设计。 在复杂的环境中, Zauner 的推测预测每个维度都存在紧凑投影 2 设计。 Pandey、 Paulsen、 Prakash 和 Rahaman 最近提议了一个方法, 以某种量子信道的缠绕断层为尺度, 使这一猜想取得数量上的进展 。 我们显示这个数量与最小的加权投影 2 设计大小相等 。 其次, 在有限的字段设置中, 我们引入了投影 2 设计的概念, 当这种投影 2 设计十分紧紧时, 我们给出了这种天体的构造 。 最后, 在四阶设置中, 我们显示每个紧凑的投影 2 设计 都决定了 D ( 2d-1) 的D ( 2d+ 1) 维度的 D ( 2d + 1) 子空间的电离子聚变聚框架 。