Non-Archimedean and p-adic Functional Welch Bounds

Abstract

We prove the non-Archimedean (resp. p-adic) Banach space version of non-Archimedean (resp. p-adic) Welch bounds recently obtained by M. Krishna. More precisely, we prove following results. 1. Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying $\left|{\sum }_{j=1}^n{\lambda }_j^2\right|={\max }_{1\le j\le n}|{\lambda }_j{|}^2$ for all ${\lambda }_j\in \mathbb{K},1\le j\le n$, for all $n\in \mathbb{N}.$ Let $\mathcal{X}$ be a $d$-dimensional non-Archimedean Banach space over $\mathbb{K}$. If $\{{\tau }_j{\}}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j{\}}_{j=1}^n$ is any collection in ${\mathcal{X}}^{\ast }$ (dual of $\mathcal{X}$) satisfying $f_j({\tau }_j)=1$ for all $1\le j\le n$ and the operator $S_{f,\tau }:{\text{Sym}}^m(\mathcal{X})\ni x\mapsto {\sum }_{j=1}^n{f}_j^{\otimes m}(x){\tau }_j^{\otimes m}\in {\text{Sym}}^m(\mathcal{X})$, is diagonalizable, then \begin{align} \text{(Non-Archimedean Functional Welch Bounds)} \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |f_j(τ_k)f_k(τ_j)|^{m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} 2. For a prime $p$, let ${\mathbb{Q}}_p$ be the p-adic number field. Let $\mathcal{X}$ be a $d$-dimensional p-adic Banach space over ${\mathbb{Q}}_p$. If $\{{\tau }_j{\}}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j{\}}_{j=1}^n$ is any collection in ${\mathcal{X}}^{\ast }$ (dual of $\mathcal{X}$) satisfying $f_j({\tau }_j)=1$ for all $1\le j\le n$ and there exists $b\in {\mathbb{Q}}_p$ such that ${\sum }_{j=1}^n{f}_j^{\otimes m}(x){\tau }_j^{\otimes m}=bx$ for all $x\in {\text{Sym}}^m(\mathcal{X}),$ then \begin{align} \text{(p-adic Functional Welch Bounds)} \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |f_j(τ_k)f_k(τ_j)|^{m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We formulate non-Archimedean functional and p-adic functional Zauner conjectures.

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