Constructing spherical designs using tight $t$-fusion frames
Ryutaro Misawa
TL;DR
The paper tackles constructing high-dimensional spherical $t$-designs by aggregating low-dimensional designs across a Grassmannian substrate using tight $t$-fusion frames. It establishes a lifting principle: given a $\mathrm{TFF}_t$ on $G_{k,d}$ and spherical $s$-designs on each subspace, one obtains a spherical $r$-design on $S^{d-1}$ with $r=\min\{s,2t+1\}$. Key contributions include a general sufficient condition for this lifting, an explicit construction of equal-weight $\mathrm{TFF}_2$ on $G_{2,d}$ via hyperoctahedral group orbits, and bounds plus nonexistence results for highly symmetric $\mathrm{ECTFF}_t$/$\mathrm{EITFF}_t$ on $G_{2,d}$, including a link to Grassmann $4$-designs and SIC-POVMs. These results provide a practical framework to synthesize high-dimensional spherical designs from structured low-dimensional inputs, offering scalable infinite families and clear symmetry limitations.
Abstract
In this paper, we study conditions under which a finite subset $Z$ of the unit sphere $S^{d-1}\subset \mathbb{R}^{d}$ becomes a spherical $t$-design, when $Z$ is constructed by the following procedure: starting from a finite set of $k$-dimensional subspaces in the real Grassmannian $G_{k,d}$, we place, for each such $k$-dimensional subspace, a finite set on its unit sphere, and then take the union of these sets in $S^{d-1}$. For this construction problem -- namely, obtaining spherical designs in higher dimensions by distributing point sets on lower-dimensional spheres subspace by subspace -- we provide a sufficient condition based on the framework of tight $t$-fusion frames ($\mathrm{TFF}_t$) due to Bachoc--Ehler. As a preparation for applications, we moreover give an explicit construction of equal-weight tight $2$-fusion frames on $G_{2,d}$ for infinitely many dimensions $d$, via unions of orbits of the hyperoctahedral group. We also derive necessary conditions for the existence of highly symmetric tight $t$-fusion frames, namely equi-chordal and equi-isoclinic tight $t$-fusion frames ($\mathrm{ECTFF}_t$ and $\mathrm{EITFF}_t$), on $G_{2,d}$, and in particular obtain bounds on the number of points.