Non-Archimedean Welch Bounds and Non-Archimedean Zauner Conjecture

Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying \begin{align*} \left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2, \quad \forall λ_j \in $\mathbb{K}$, 1\leq j \leq n, \forall n \in $\mathbb{N}$. \end{align*} For $d\in \mathbb{N}$, let ${\mathbb{K}}^d$ be the standard $d$-dimensional non-Archimedean Hilbert space. Let $m\in \mathbb{N}$ and ${\text{Sym}}^m({\mathbb{K}}^d)$ be the non-Archimedean Hilbert space of symmetric m-tensors. We prove the following result. If $\{{\tau }_j{\}}_{j=1}^n$ is a collection in ${\mathbb{K}}^d$ satisfying $⟨{\tau }_j,{\tau }_j⟩=1$ for all $1\le j\le n$ and the operator ${\text{Sym}}^m({\mathbb{K}}^d)\ni x\mapsto {\sum }_{j=1}^n⟨x,{\tau }_j^{\otimes m}⟩{\tau }_j^{\otimes m}\in {\text{Sym}}^m({\mathbb{K}}^d)$ is diagonalizable, then \begin{align} (1) \quad \quad \quad \max_{1\leq j,k \leq n, j \neq k}{|n|, |\langle τ_j, τ_k\rangle|^{2m} }\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We call Inequality (1) as the non-Archimedean version of Welch bounds obtained by Welch [$\mathit{\text{IEEE}}\mathit{\text{Transactions}}\mathit{\text{on}}\mathit{\text{Information}}\mathit{\text{Theory,}}\mathit{\text{1974}}$]. We formulate non-Archimedean Zauner conjecture.