Equiangular lines via matrix projection
Igor Balla
TL;DR
This work introduces a unifying projection-based framework for bounding the size of equiangular line systems in real and complex spaces by orthogonally projecting Gram-type matrices onto spans generated by rank-1 projections. The method yields sharp, regime-aware bounds that bridge previously separate regimes, extends the Alon–Boppana theorem to dense graphs, and provides refined Welch-type bounds. In the real and complex settings, the authors obtain tight or near-tight bounds and connect these to strongly regular graphs, SIC-POVM structures, and Zauner-type conjectures. The results offer a powerful spectral-graph-theoretic toolkit with potential applications in quantum information and coding theory, and they suggest promising avenues for further generalizations to higher-order codes and signed graphs.
Abstract
In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in $\mathbb{R}^r$ with angle $\arccos(α)$ and gave a partial answer in the regime $r \leq 1/α^2 - 2$. At the other extreme where $r$ is at least exponential in $1/α$, recent breakthroughs have led to an almost complete resolution of this problem. In this paper, we introduce a new method for obtaining upper bounds which unifies and improves upon previous approaches, thereby yielding bounds which bridge the gap between the aforementioned regimes and are best possible either exactly or up to a small multiplicative constant. Our approach relies on orthogonal projection of matrices with respect to the Frobenius inner product and as a byproduct, it yields the first extension of the Alon-Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to $\binom{r+1}{2}$ equiangular lines in $\mathbb{R}^r$. Applications of our method in the complex setting will be discussed as well.