论文标题
SICS的维度塔。 ii。一些结构
Dimension towers of SICs. II. Some constructions
论文作者
论文摘要
SIC是有限的尺寸希尔伯特空间中最大的等缘紧密框架。鉴于尺寸$ d $的SIC，有充分的证据表明，始终存在一个尺寸$ d（d-2）$的对齐的SIC，具有可预测的对称性和嵌入其中的较小的equiangular紧密帧。我们提供了如何计算共享这些属性的dimension $ d（d-2）$中的向量集的配方。它们由在输入$ d $ dimensional sic的数字定义的某些子空间中的最大纠结向量组成。但是，构造包含免费参数，我们尚未证明可以始终选择它们，以便其中一组向量是SIC。我们提供一些工作的例子，我们希望可以向读者建议如何改善我们的施工。为简单起见，我们将自己限制在奇数维度的情况下。
A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension $d$, there is good evidence that there always exists an aligned SIC in dimension $d(d-2)$, having predictable symmetries and smaller equiangular tight frames embedded in them. We provide a recipe for how to calculate sets of vectors in dimension $d(d-2)$ that share these properties. They consist of maximally entangled vectors in certain subspaces defined by the numbers entering the $d$ dimensional SIC. However, the construction contains free parameters and we have not proven that they can always be chosen so that one of these sets of vectors is a SIC. We give some worked examples that, we hope, may suggest to the reader how our construction can be improved. For simplicity we restrict ourselves to the case of odd dimensions.
