Noncommutative Equiangular Lines: van Lint-Seidel Relative and Gerzon Universal Bounds

1. Introduction

Let $d\in \mathbb{N}$ and $\gamma \in [0,1]$. Recall that a collection ${\left\{{\tau }_j\right\}}_{j=1}^n$ of unit vectors in ${\mathbb{R}}^d$ or ${\mathbb{C}}^d$ is said to be $\gamma$-equiangular lines [1,2] if

$$\begin{array}{c}\hfill |\langle {\tau }_j,{\tau }_k\rangle |=\gamma ,\forall 1\le j,k\le n,j\ne k.\end{array}$$

Two fundamental problems associated with equiangular lines are the following.

Problem 1. Given $d\in \mathbb{N}$ and $\gamma \in [0,1]$, what is the upper bound on n such that there exists a collection ${\left\{{\tau }_j\right\}}_{j=1}^n$ of $\gamma$-equiangular lines in ${\mathbb{R}}^d$?

Problem 2. For which parameters $d,n\in \mathbb{N}$ and $\gamma \in [0,1]$, there exists a collection ${\left\{{\tau }_j\right\}}_{j=1}^n$ of $\gamma$-equiangular lines in ${\mathbb{R}}^d$? If exists, how to construct them?

Two answers to Problem (1) which are fundamental driving force in the study of equiangular lines are the following results of van Lint and Seidel [1,3] and Gerzon [4].

Theorem 1.

[1,3,5,6] (van Lint-Seidel Relative Bound) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be γ-equiangular lines in ${\mathbb{R}}^d$ or ${\mathbb{C}}^d$. Then

$$\begin{array}{c}\hfill n(1-d{\gamma }^2)\le d(1-{\gamma }^2).\end{array}$$

In particular, if

$$\begin{array}{c}\hfill \gamma <{\displaystyle \frac{1}{\sqrt{d}}},\end{array}$$

then

$$\begin{array}{c}\hfill n\le {\displaystyle \frac{d(1-{\gamma }^2)}{1-d{\gamma }^2}}.\end{array}$$

(1)

Theorem 2.

[4] (Gerzon Universal Bound)

Theorem 3.

((i)).

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be γ-equiangular lines in ${\mathbb{R}}^d$. Then

$$\begin{array}{c}\hfill n\le {\displaystyle \frac{d(d+1)}{2}}.\end{array}$$

(2)

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be γ-equiangular lines in ${\mathbb{C}}^d$. Then

$$\begin{array}{c}\hfill n\le d^2.\end{array}$$

(3)

Bounds in Inequalities (1), (2) and (3) are improved for various special values of $\gamma$, n and d (including asymptotic). Regarding Problem (2), various special cases are solved but full generality remains open. We refer [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] for a look on these achievements in the real case and [31,32,33,34,35,36,37] in the complex case. In the real case, it is known that equality is not always attained for every d in Inequality (2). Celebrated Zauner conjecture says that there exists a $d^2$${\displaystyle \frac{1}{\sqrt{d+1}}}$-equiangular lines in ${\mathbb{C}}^d$ for every d satisfying equality in Inequality (3) [38,39,40,41,42,43,44].

We wish to record that there is a vector space version of equiangular lines [45,46] and there is also a notion of equiangular subspaces of Euclidean spaces [47,48]. Recently, p-adic version of equiangular lines have been introduced [49].

As noncommutative geometry plays major role in modern Mathematics, our fundamental objective is to initiate a study of noncommutative equiangular lines. As most important results in the theory of equiangular lines are van Lint-Seidel relative and Gerzon universal bounds, we derive noncommutative versions of them.