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}\lambda_j^2\right|=\max_{1\leq j \leq n}|\lambda_j|^2, \quad \forall \lambda_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 $\langle \tau_j, \tau_j\rangle =1$ for all $1\leq j \leq n$ and the operator $\text{Sym}^m(\mathbb{K}^d)\ni x \mapsto \sum_{j=1}^n\langle x, \tau_j^{\otimes m}\rangle \tau_j^{\otimes m} \in \text{Sym}^m(\mathbb{K}^d)$ 
 is diagonalizable, then 
 \begin{align}\label{WELCHNONABSTRACT}
          \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{align}
We call Inequality (\ref{WELCHNONABSTRACT})   as the non-Archimedean version of Welch bounds obtained by     Welch [\textit{IEEE Transactions on  Information Theory, 1974}].  We formulate non-Archimedean  Zauner conjecture.