$p$-Adic Welch Bounds and $p$-Adic Zauner Conjecture

Abstract

Let $p$ be a prime. For $d\in \mathbb{N}$, let $\mathbb{Q}_p^d$ be the standard $d$-dimensional p-adic Hilbert space. Let $m \in \mathbb{N}$ and $\text{Sym}^m(\mathbb{Q}_p^d)$ be the $p$-adic Hilbert space of symmetric m-tensors. We prove the following result. Let $\{\tau_j\}_{j=1}^n$ be a collection in $\mathbb{Q}_p^d$ satisfying (i) $\langle \tau_j, \tau_j\rangle =1$ for all $1\leq j \leq n$ and (ii) there exists $b \in \mathbb{Q}_p$ satisfying $\sum_{j=1}^{n}\langle x, \tau_j\rangle \tau_j =bx$ for all $x \in \mathbb{Q}^d_p.$ Then

$$\begin{aligned} \, \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle \tau_j, \tau_k\rangle|^{2m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{aligned}$$

(0.1)

We call Inequality (0.1) as the $p$-adic version of Welch bounds obtained by Welch [IEEE Transactions on Information Theory, 1974]. Inequality (0.1) differs from the non-Archimedean Welch bound obtained recently by M. Krishna as one can not derive one from another. We formulate $p$-adic Zauner conjecture.

1. Introduction

In 1974 Prof. L. Welch proved the following result [91].

Theorem 1.1.

[91] (Welch Bounds) Let $n> d$. If $\{\tau_j\}_{j=1}^n$ is any collection of unit vectors in $\mathbb{C}^d$, then

$$\begin{aligned} \, \sum_{1\leq j,k \leq n}|\langle \tau_j, \tau_k\rangle |^{2m}=\sum_{j=1}^n\sum_{k=1}^n|\langle \tau_j, \tau_k\rangle |^{2m}\geq \frac{n^2}{{d+m-1\choose m}}, \quad \forall m \in \mathbb{N}. \end{aligned}$$

In particular,

$$\begin{aligned} \, \sum_{1\leq j,k \leq n}|\langle \tau_j, \tau_k\rangle |^{2}= \sum_{j=1}^n\sum_{k=1}^n|\langle \tau_j, \tau_k\rangle |^{2}\geq \frac{n^2}{{d}}. \end{aligned}$$

Further,

$$\begin{aligned} \, \boldsymbol{(Higher\ order\ Welch\ bounds)} \quad \max _{1\leq j,k \leq n, j\neq k}|\langle \tau_j, \tau_k\rangle |^{2m}\geq \frac{1}{n-1}\left[\frac{n}{{d+m-1\choose m}}-1\right], \quad \forall m \in \mathbb{N}. \end{aligned}$$

In particular,

$$\begin{aligned} \, \boldsymbol{(First\ order\ Welch\ bound)}\quad \max _{1\leq j,k \leq n, j\neq k}|\langle \tau_j, \tau_k\rangle |^{2}\geq\frac{n-d}{d(n-1)}. \end{aligned}$$

It is impossible to list all applications of Theorem 1.1. A few are: in the study of root-mean-square (RMS) absolute cross relation of unit vectors [75], frame potential [10, 20, 15], correlations [74], codebooks [31], numerical search algorithms [93, 92], quantum measurements [77], coding and communications [86, 81], code division multiple access (CDMA) systems [55, 56], wireless systems [72], compressed/compressive sensing [84, 87, 36, 33, 7, 85, 76, 2], ‘game of Sloanes’ [49], equiangular tight frames [82], equiangular lines [65, 25, 35, 48], digital fingerprinting [64] etc.

Theorem 1.1 has been improved/different proofs were given in [21, 27, 89, 88, 26, 73, 46, 32, 81]. In 2021 M. Krishna derived continuous version of Theorem 1.1 [59]. In 2022 M. Krishna obtained Theorem 1.1 for Hilbert C*-modules [57], Banach spaces [60] and non-Archimedean Hilbert spaces [58].

In this paper we derive $p$-adic Welch bounds (Theorem 2.5). We formulate $p$-adic Zauner conjecture (Conjecture 3.3).

Motivation: Following are some of the important connections of Welch bounds to other prominent areas of research which made us to consider Welch bounds in $p$-adic setting.

• (I) Spherical $t$-designs have direct relation with Welch bounds (for instance, see Chapter 6 in [90]). Existence of spherical $t$-designs is known (see [79]) but their exact number is not known. Asymptotic bounds for spherical $t$-design are recently derived (see [16]).

• (II) Welch bounds are essential in the study of equiangular lines (see Chapter 12 in [90]). Existence of equiangular lines having a prescribed angle in a given dimension is largely unknown (see [83]). Asymptotic bound for equiangular lines is recently derived (see [51]).

• (III) Benedetto and Fickus (see [10]) characterized finite unit norm frames for finite dimensional Hilbert spaces using frame potential which has connection with Welch bounds (see Chapter 6 in [90]). This characterization motivated the development of so called Fundamental Inequality for Finite Frames (see [20]).

• (IV) In compressive sensing, Welch bounds are required in the construction of matrices with small coherence which uses the inner product (see Chapter 5 in [36]).

• (V) An easy observation associated with first order Welch bound is that it gives van Lint-Seidel relative bound for equiangular lines which was obtained by van-Lint and Seidel in a different method (without knowing Welch bounds) [62]. Equiangular lines have interaction with several areas which suggests the use of Welch bounds in different areas.

• (VI) Recently it is noticed that using Welch bounds one can show that Johnson-Lindenstrauss lemma is optimal [61]. As it is well-known, Johnson-Lindenstrauss lemma has applications even in computer science. This makes the way of Welch bounds to computer science.

2. $p$-adic Welch bounds

We begin by recalling the notion of $p$-adic Hilbert space. We refer [52, 29, 30, 53, 1] for more on $p$-adic Hilbert spaces.

Definition 2.1.

[29, 30] Let $\mathbb{K}$ be a non-Archimedean valued field (with valuation $|\cdot|$) and $\mathcal{X}$ be a non-Archimedean Banach space (with norm $\|\cdot\|$) over $\mathbb{K}$. We say that $\mathcal{X}$ is a $p$-adic Hilbert space if there is a map (called as $p$-adic inner product) $\langle \cdot, \cdot \rangle: \mathcal{X} \times \mathcal{X} \to \mathbb{K}$ satisfying following.

• (i) If $x \in \mathcal{X}$ is such that $\langle x,y \rangle =0$ for all $y \in \mathcal{X}$, then $x=0$.

• (ii) $\langle x, y \rangle =\langle y, x \rangle$ for all $x,y \in \mathcal{X}$.

• (iii) $\langle \alpha x, y+z \rangle =\alpha \langle x, y \rangle+\langle x,z\rangle$ for all $\alpha \in \mathbb{K}$, for all $x,y,z \in \mathcal{X}$.

• (iv) $|\langle x, y \rangle |\leq \|x\|\|y\|$ for all $x,y \in \mathcal{X}$.

Following is the standard example which we consider in the paper.

Example 2.2.

[52] Let $p$ be a prime. For $d \in \mathbb{N}$, let $\mathbb{Q}_p^d$ be the standard p-adic Hilbert space equipped with the inner product

$$\begin{aligned} \, \langle (a_j)_{j=1}^d,(b_j)_{j=1}^d\rangle := \sum_{j=1}^da_jb_j, \quad \forall (a_j)_{j=1}^d,(b_j)_{j=1}^d \in \mathbb{Q}_p^d \end{aligned}$$

and norm

$$\begin{aligned} \, \|(x_j)_{j=1}^d\|:= \max_{1\leq j \leq d}|x_j|, \quad \forall (x_j)_{j=1}^d\in \mathbb{Q}_p^d. \end{aligned}$$

Let $I_{\mathbb{Q}_p^d}$ be the identity operator on $\mathbb{Q}_p^d$. Note that $\mathbb{Q}_p^d$ is not a non-Archimedean Hilbert space as it does not satisfies Equation (2) in [58] (see Page 40, [69]). Following is the first important result of the paper.

Theorem 2.3.

(First Order $p$-adic Welch Bound) Let $p$ be a prime and $n, d \in \mathbb{N}$. If $\{\tau_j\}_{j=1}^n$ is any collection in $\mathbb{Q}_p^d$ such that there exists $b \in \mathbb{Q}_p$ satisfying

$$\begin{aligned} \, \sum_{j=1}^{n}\langle x, \tau_j\rangle \tau_j =bx, \quad \forall x \in \mathbb{Q}^d_p, \end{aligned}$$

then

$$\begin{aligned} \, \max_{1\leq j,k \leq n, j \neq k}\left \{\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^2 \right|, |\langle \tau_j,\tau_k\rangle|^2\right\}\geq \frac{1}{|d|} \left|\sum_{j=1}^n\langle \tau_j, \tau_j \rangle \right|^2. \end{aligned}$$

In particular, if $\langle \tau_j, \tau_j\rangle =1$ for all $1\leq j \leq n$, then

$$\begin{aligned} \, \boldsymbol{(First\ order\ }p\textbf{-}\boldsymbol{adic\ Welch\ bound)} \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle \tau_j, \tau_k\rangle|^2 \}\geq \frac{|n|^2}{|d|}. \end{aligned}$$

Proof. Define $S_\tau : \mathbb{Q}_p^d\ni x \mapsto \sum_{j=1}^n\langle x, \tau_j\rangle \tau_j \in \mathbb{Q}_p^d$. Then

$$\begin{aligned} \, &bd=\operatorname{Tra}(bI_{\mathbb{Q}_p^d})=\operatorname{Tra}(S_{\tau})=\sum_{j=1}^n\langle \tau_j, \tau_j \rangle , \\ & b^2d=\operatorname{Tra}(b^2I_{\mathbb{Q}_p^d})=\operatorname{Tra}(S^2_{\tau})=\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j, \tau_k \rangle\langle \tau_k, \tau_j \rangle. \end{aligned}$$

Therefore

$$\begin{aligned} \, \left|\sum_{j=1}^n\langle \tau_j, \tau_j \rangle \right|^2&=|\operatorname{Tra}(S_{\tau})|^2=|bd|^2=|d||b^2d| =|d|\left|\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j, \tau_k \rangle\langle \tau_k, \tau_j \rangle\right|\\ &=|d|\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^2+\sum_{j,k=1, j\neq k}^n\langle \tau_j,\tau_k\rangle \langle \tau_k,\tau_j\rangle\right|\\ &\leq |d|\max \left\{\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^2 \right|, \left|\sum_{j,k=1, j\neq k}^n\langle \tau_j,\tau_k\rangle \langle \tau_k,\tau_j\rangle\right|\right\}\\ &\leq |d|\max \left\{\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^2 \right|, \max_{1\leq j,k \leq n, j \neq k}|\langle \tau_j,\tau_k\rangle \langle \tau_k,\tau_j\rangle|\right\}\\ &= |d| \max_{1\leq j,k \leq n, j \neq k}\left \{\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^2 \right|, |\langle \tau_j,\tau_k\rangle \langle \tau_k,\tau_j\rangle| \right\}\\ &= |d| \max_{1\leq j,k \leq n, j \neq k}\left \{\left| \sum_{l=1}^n\langle \tau_l,\tau_l\rangle^2 \right|, |\langle \tau_j,\tau_k\rangle|^2\right\}. \end{aligned}$$

Whenever $\langle \tau_j, \tau_j\rangle =1$ for all $1\leq j \leq n$,

$$\begin{aligned} \, |n|^2\leq |d|\max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle \tau_j, \tau_k\rangle|^2 \}. \end{aligned}$$

We next derive higher order $p$-adic Welch bounds. For this we need the concept of vector space of symmetric tensors. Given a vector space $\mathcal{V}$ of dimension $d$, let $\mathcal{V}^{\otimes m}$ be the vector space of $m$-tensors. A vector

$$\begin{aligned} \, \sum_{j=1}^{n}x_{j,1}\otimes \cdots \otimes x_{j,m} \in \mathcal{V}^{\otimes m} \end{aligned}$$

is said to be symmetric if for every bijection $\sigma:\{1, \dots, m\}\to \{1, \dots, m\}$, we have

$$\begin{aligned} \, \sum_{j=1}^{n}x_{j,\sigma(1)}\otimes \cdots \otimes x_{j,\sigma(m)} =\sum_{j=1}^{n}x_{j,1}\otimes \cdots \otimes x_{j,m}. \end{aligned}$$

Set of all symmetric $m$-tensors will form a vector space, denoted by $\text{Sym}^m(\mathcal{V})$. Following result will give dimension of this space.

Theorem 2.4.

[23, 14] If $\mathcal{V}$ is a vector space of dimension $d$ and $\text{Sym}^m(\mathcal{V})$ denotes the vector space of symmetric m-tensors, then

$$\begin{aligned} \, \textit{dim}(\textit{Sym}^m(\mathcal{V}))={d+m-1 \choose m}, \quad \forall m \in \mathbb{N}. \end{aligned}$$

Theorem 2.5.

(Higher Order $p$-adic Welch Bounds) Let $p$ be a prime and $n, d, m \in \mathbb{N}$. If $\{\tau_j\}_{j=1}^n$ is any collection in $\mathbb{Q}_p^d$ such that there exists $b \in \mathbb{Q}_p$ satisfying

$$\begin{aligned} \, \sum_{j=1}^{n}\langle x, \tau_j^{\otimes m}\rangle \tau_j^{\otimes m} =bx, \quad \forall x \in \textit{Sym}^m(\mathbb{Q}_p^d), \end{aligned}$$

then

$$\begin{aligned} \, \max_{1\leq j,k \leq n, j \neq k}\left \{\left| \sum_{l=1}^n\langle \tau_l,\tau_l\rangle^{2m} \right|, |\langle \tau_j,\tau_k\rangle|^{2m}\right\}\geq \frac{1}{\left|{d+m-1 \choose m}\right|}\left|\sum_{j=1}^n\langle \tau_j, \tau_j \rangle^m \right|^2. \end{aligned}$$

In particular, if $\langle \tau_j, \tau_j\rangle =1$ for all $1\leq j \leq n$, then

$$\begin{aligned} \, \boldsymbol{(Higher\ order\ }p\textbf{-}\boldsymbol{adic\ Welch\ bound)} \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle \tau_j, \tau_k\rangle|^{2m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{aligned}$$

Proof. Define $S_\tau : \text{Sym}^m(\mathbb{Q}_p^d)\ni x \mapsto \sum_{j=1}^n\langle x, \tau_j^{\otimes m}\rangle \tau_j^{\otimes m} \in \text{Sym}^m(\mathbb{Q}_p^d)$. Then

$$\begin{aligned} \, &b\operatorname{dim(\text{Sym}^m(\mathbb{Q}_p^d))}=\operatorname{Tra}(bI_{\text{Sym}^m(\mathbb{Q}_p^d)})=\operatorname{Tra}(S_{\tau})=\sum_{j=1}^n\langle \tau_j^{\otimes m} , \tau_j^{\otimes m} \rangle , \\ & b^2\operatorname{dim(\text{Sym}^m(\mathbb{Q}_p^d))}=\operatorname{Tra}(b^2I_{\text{Sym}^m(\mathbb{Q}_p^d)})=\operatorname{Tra}(S^2_{\tau})=\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j^{\otimes m} , \tau^{\otimes m} _k \rangle\langle \tau^{\otimes m} _k, \tau^{\otimes m} _j \rangle. \end{aligned}$$

Therefore by using Theorem 2.4 we get

$$\begin{aligned} \, &\left|\sum_{j=1}^n\langle \tau_j, \tau_j \rangle^m \right|^2= \left|\sum_{j=1}^n\langle \tau_j^{\otimes m}, \tau_j^{\otimes m} \rangle \right|^2=|\operatorname{Tra}(S_{\tau})|^2=\left|b\operatorname{dim(\text{Sym}^m(\mathbb{Q}_p^d))}\right|^2\\ &=\left|\operatorname{dim(\text{Sym}^m(\mathbb{Q}_p^d))}\right|\left|b^2\operatorname{dim(\text{Sym}^m(\mathbb{Q}_p^d))}\right|\\ &=\left|\operatorname{dim(\text{Sym}^m(\mathbb{Q}_p^d))}\right|\left|\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j^{\otimes m}, \tau_k^{\otimes m} \rangle\langle \tau_k^{\otimes m}, \tau_j^{\otimes m} \rangle\right|\\ &=\left|{d+m-1 \choose m}\right|\left|\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j^{\otimes m}, \tau_k^{\otimes m} \rangle\langle \tau_k^{\otimes m}, \tau_j^{\otimes m} \rangle\right|\\ &=\left|{d+m-1 \choose m}\right|\left|\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j, \tau_k \rangle^m\langle \tau_k, \tau_j \rangle^m\right|\\ &=\left|{d+m-1 \choose m}\right|\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^{2m}+\sum_{j,k=1, j\neq k}^n\langle \tau_j,\tau_k\rangle^m \langle \tau_k,\tau_j\rangle^m\right|\\ &\leq \left|{d+m-1 \choose m}\right|\max \left\{\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^{2m} \right|, \left|\sum_{j,k=1, j\neq k}^n\langle \tau_j,\tau_k\rangle^m \langle \tau_k,\tau_j\rangle^m\right|\right\}\\ &\leq \left|{d+m-1 \choose m}\right|\max \left\{\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^{2m} \right|, \max_{1\leq j,k \leq n, j \neq k}|\langle \tau_j,\tau_k\rangle^m \langle \tau_k,\tau_j\rangle^m|\right\}\\ &\leq \left|{d+m-1 \choose m}\right| \max_{1\leq j,k \leq n, j \neq k}\left \{\left| \sum_{l=1}^n\langle \tau_l,\tau_l \rangle^{2m} \right|, |\langle \tau_j,\tau_k\rangle^m \langle \tau_k,\tau_j\rangle^m| \right\}\\ &= \left|{d+m-1 \choose m}\right| \max_{1\leq j,k \leq n, j \neq k}\left \{\left| \sum_{l=1}^n\langle \tau_l,\tau_l\rangle^{2m} \right|, |\langle \tau_j,\tau_k\rangle|^{2m}\right\}. \end{aligned}$$

Whenever $\langle \tau_j, \tau_j\rangle =1$ for all $1\leq j \leq n$,

$$\begin{aligned} \, |n|^2\leq \left|{d+m-1 \choose m}\right| \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle \tau_j, \tau_k\rangle|^{2m} \}. \end{aligned}$$

Remark 2.6.

Conditions given in the Theorem 2.5 says that the operator $S_\tau$ in the proof of Theorem 2.5 is diagonalizable. Thus Theorem 2.5 is restrictive as the hypothesis is stronger than that of Theorem 2.3 in [58]. However, note that the field $\mathbb{Q}_p$ does not satisfies the Equation (2) in [58] (see [69]) and hence neither the results in this paper can be derived from the results in [58] nor the results in [58] can be derived from the results in this paper.

Remark 2.7.

Theorems 2.3 and 2.5 hold by replacing $\mathbb{Q}_p^d$ by a $d$-dimensional p-adic Hilbert space over any non-Archimedean (complete) valued field (such as $\mathbb{C}_p$).

3. $p$-adic Zauner Conjecture and open problems

Using Theorem 2.3 we ask the following question.

Question 3.1.

Given a prime $p$, for which $(d,n) \in \mathbb{N}\times \mathbb{N}$, there exist vectors $\tau_1, \dots, \tau_n \in \mathbb{Q}_p^d$ satisfying the following.

• (i) $\langle \tau_j, \tau_j \rangle =1$ for all $1\leq j \leq n$.

• (ii) There exists $b \in \mathbb{Q}_p$ satisfying

  $$\begin{aligned} \, \sum_{j=1}^{n}\langle x, \tau_j\rangle \tau_j =bx, \quad \forall x \in \mathbb{Q}^d_p. \end{aligned}$$
• (iii)
  $$\begin{aligned} \, \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle \tau_j, \tau_k\rangle|^2 \}= \frac{|n|^2}{|d|}. \end{aligned}$$

We can formulate a strong form of Question 3.1 as follows.

Question 3.2.

Given a prime $p$, for which $(d,n) \in \mathbb{N}\times \mathbb{N}$, there exist vectors $\tau_1, \dots, \tau_n \in \mathbb{Q}_p^d$ satisfying the following.

• (i) $\langle \tau_j, \tau_j \rangle =1$ for all $1\leq j \leq n$.

• (ii) There exists $b \in \mathbb{Q}_p$ satisfying

  $$\begin{aligned} \, \sum_{j=1}^{n}\langle x, \tau_j\rangle \tau_j =bx, \quad \forall x \in \mathbb{Q}^d_p. \end{aligned}$$
• (iii)
  $$\begin{aligned} \, \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle \tau_j, \tau_k\rangle|^2 \}= \frac{|n|^2}{|d|}. \end{aligned}$$

• (iv) $\|\tau_j\| =1$ for all $1\leq j \leq n$.

Why Question 3.2 is different than Question 3.1? Reason is that unlike non-Archimedean Hilbert spaces, in $p$-adic Hilbert spaces, norm is not defined as $\sqrt{|\langle \cdot, \cdot\rangle|}.$ A particular case of Question 3.1 is the following p-adic version of Zauner conjecture (see [6, 4, 95, 78, 37, 71, 3, 12, 5, 54, 40, 13, 47, 11, 63, 59] for Zauner conjecture in Hilbert spaces, [57] for Zauner conjecture in Hilbert C*-modules, [60] for Zauner conjecture in Banach spaces and [58] for Zauner conjecture in non-Archimedean Hilbert spaces).

Conjecture 3.3.

($p$-adic Zauner Conjecture) Let $p$ be a prime. For each $d\in \mathbb{N}$, there exist vectors $\tau_1, \dots, \tau_{d^2} \in \mathbb{Q}_p^d$ satisfying the following.

• (i) $\langle \tau_j, \tau_j \rangle =1$ for all $1\leq j \leq d^2$.

• (ii) There exists $b \in \mathbb{Q}_p$ satisfying

  $$\begin{aligned} \, \sum_{j=1}^{d^2}\langle x, \tau_j\rangle \tau_j =bx, \quad \forall x \in \mathbb{Q}^d_p. \end{aligned}$$
• (iii)
  $$\begin{aligned} \, |\langle \tau_j, \tau_k\rangle|^2 =|d|, \quad \forall 1\leq j, k \leq d^2, j \neq k. \end{aligned}$$

Question 3.2 gives the following Zauner conjecture.

Conjecture 3.4.

($p$-adic Zauner Conjecture - strong form) Let $p$ be a prime. For each $d\in \mathbb{N}$, there exist vectors $\tau_1, \dots, \tau_{d^2} \in \mathbb{Q}_p^d$ satisfying the following.

• (i) $\langle \tau_j, \tau_j \rangle =1$ for all $1\leq j \leq d^2$.

• (ii) There exists $b \in \mathbb{Q}_p$ satisfying

  $$\begin{aligned} \, \sum_{j=1}^{d^2}\langle x, \tau_j\rangle \tau_j =bx, \quad \forall x \in \mathbb{Q}^d_p. \end{aligned}$$
• (iii)
  $$\begin{aligned} \, |\langle \tau_j, \tau_k\rangle|^2 =|d|, \quad \forall 1\leq j, k \leq d^2, j \neq k. \end{aligned}$$

• (iv) $\|\tau_j\| =1$ for all $1\leq j \leq d^2$.

We recall the definition of Gerzon’s bound which allows us to remember companions to Welch bounds in Hilbert spaces.

Definition 3.5.

[49] Given $d\in \mathbb{N}$, define Gerzon’s bound

$$\begin{aligned} \, \mathcal{Z}(d, \mathbb{K}):= \left\{ \begin{array}{cc} d^2 & \quad \text{if} \quad \mathbb{K} =\mathbb{C}\\ \frac{d(d+1)}{2} & \quad \text{if} \quad \mathbb{K} =\mathbb{R}.\\ \end{array} \right. \end{aligned}$$

Theorem 3.6.

[49, 92, 66, 80, 18, 24, 45, 70] Define $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $m:= \operatorname{dim}_{\mathbb{R}}(\mathbb{K})/2$. If $\{\tau_j\}_{j=1}^n$ is any collection of unit vectors in $\mathbb{K}^d$, then

• (i) (Bukh-Cox bound)

  $$\begin{aligned} \, \max _{1\leq j,k \leq n, j\neq k}|\langle \tau_j, \tau_k\rangle |\geq \frac{\mathcal{Z}(n-d, \mathbb{K})}{n(1+m(n-d-1)\sqrt{m^{-1}+n-d})-\mathcal{Z}(n-d, \mathbb{K})}\quad \textit{if} \quad n>d. \end{aligned}$$

• (ii) (Orthoplex/Rankin bound)

  $$\begin{aligned} \, \max _{1\leq j,k \leq n, j\neq k}|\langle \tau_j, \tau_k\rangle |\geq\frac{1}{\sqrt{d}} \quad \textit{if} \quad n>\mathcal{Z}(d, \mathbb{K}). \end{aligned}$$

• (iii) (Levenstein bound)

  $$\begin{aligned} \, \max _{1\leq j,k \leq n, j\neq k}|\langle \tau_j, \tau_k\rangle |\geq \sqrt{\frac{n(m+1)-d(md+1)}{(n-d)(md+1)}} \quad \textit{if} \quad n>\mathcal{Z}(d, \mathbb{K}). \end{aligned}$$

• (iv) (Exponential bound)

  $$\begin{aligned} \, \max _{1\leq j,k \leq n, j\neq k}|\langle \tau_j, \tau_k\rangle |\geq 1-2n^{\frac{-1}{d-1}}. \end{aligned}$$

Theorem 3.6 and Theorem 2.3 give the following problem.

Question 3.7.

Whether there is a $p$-adic version of Theorem 3.6? In particular, does there exists a version of

• (i) $p$-adic Bukh-Cox bound?

• (ii) $p$-adic Orthoplex/Rankin bound?

• (iii) $p$-adic Levenstein bound?

• (iv) $p$-adic Exponential bound?

We already wrote that Welch bounds have applications in study of equiangular lines. We wish to formulate equiangular line problem for $p$-adic Hilbert spaces. For the study of equiangular lines in Hilbert spaces we refer [62, 51, 41, 8, 17, 28, 38, 9, 50, 67, 44, 39, 19, 68, 94], quaternion Hilbert spaces we refer [34], octonion Hilbert spaces we refer [22], finite dimensional vector spaces over finite fields we refer [42, 43], for Banach spaces we refer [60] and for non-Archimedean Hilbert spaces we refer [58].

Question 3.8.

($p$-adic Equiangular Line Problem) Let $p$ be a prime. Given $a\in \mathbb{Q}_p$, $d \in \mathbb{N}$ and $\gamma>0$, what is the maximum $n =n(p, a,d, \gamma)\in \mathbb{N}$ such that there exist vectors $\tau_1, \dots, \tau_n \in \mathbb{Q}_p^d$ satisfying the following.

• (i) $\langle \tau_j, \tau_j \rangle =a$ for all $1\leq j \leq n$.

• (ii) $|\langle \tau_j, \tau_k \rangle|^2 =\gamma$ for all $1\leq j, k \leq n, j \neq k$.

In particular, whether there is a $p$-adic Gerzon bound?

Question 3.8 can be easily lifted to formulate question of $p$-adic regular $s$-distance sets.