p-adic Welch Bounds and p-adic Zauner Conjecture

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) $⟨{\tau }_j,{\tau }_j⟩=1$ for all $1\le j\le n$ and (ii) there exists $b\in {\mathbb{Q}}_p$ satisfying ${\sum }_{j=1}^n⟨x,{\tau }_j⟩{\tau }_j=bx$ for all $x\in {\mathbb{Q}}_p^d\mathrm{.}$ 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 p-adic version of Welch bounds obtained by Welch [$\mathit{\text{IEEE}}\mathit{\text{Transactions}}\mathit{\text{on}}\mathit{\text{Information}}\mathit{\text{Theory,}}\mathit{\text{1974}}$]. Inequality (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.