# Nearly orthogonal vectors and small antipodal spherical codes

## TL;DR

This paper investigates how to arrange $d+k$ vectors in $\\mathbb{R}^d$ to be as close to orthogonal as possible, establishing bounds and connections to equiangular lines, with implications for complex spaces and quantum physics.

## Contribution

The paper derives bounds on the minimal maximum inner product for nearly orthogonal vectors and links these bounds to the existence of equiangular line systems in real and complex spaces.

## Key findings

- Exact values of $ heta(d,k)$ for specific small $k$
- Asymptotic bounds for large $d$ and fixed $k$
- Connection to equiangular lines and SIC-POVM in quantum physics

## Abstract

How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $\theta(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $\theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $\theta(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establish asymptotics for general $k$. The main tool is an upper bound on $\mathbb{E}_{x,y\sim\mu}|\langle x,y\rangle|$ whenever $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, which may be of independent interest. Our results translate naturally to the analogous question in $\mathbb{C}^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $\mathbb{C}^k$, also known as SIC-POVM in physics literature.