$$p$$ -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.