A note on tight projective 2‐designs
A note on tight projective 2‐designs
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DOI：
10.1002/jcd.21804
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发表时间：
2021-01
影响因子：
0.7
通讯作者：
Joseph W. Iverson;E. King;D. Mixon
中科院分区：
文献类型：
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作者：
Joseph W. Iverson;E. King;D. Mixon
我们在三个不同的背景下研究了紧射影2-设计。在复杂背景下，Zauner猜想预言了在每个维度上都存在紧射影2-设计。Pandey、Paulsen、Prakash和Rahaman最近提出了一种方法，根据某个量子通道的纠缠破缺秩数对这一猜想进行量化改进。我们证明了这个量等于最小加权射影2-设计的大小。其次，在有限域环境下，我们引入了射影2-设计的概念，刻划了这种射影2-设计是紧的，并给出了这类对象的一个构造。最后，在四元数情形下，证明了Hd的每个紧射影2-设计决定了3维Rd(2 d+1)的d(2 d−1)子空间的等同宿紧融合框架.
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 ( 2 d − 1 ) subspaces of R d ( 2 d + 1 ) of dimension 3.