p-adic Welch Bounds and p-adic Zauner Conjecture
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916347368177664 |
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| author | Krishna, K. Mahesh |
| author_facet | Krishna, K. Mahesh |
| contents | 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 $\{τ_j\}_{j=1}^n$ be a collection in $\mathbb{Q}_p^d$ satisfying (i) $\langle τ_j, τ_j\rangle =1$ for all $1\leq j \leq n$ and (ii) there exists $b \in \mathbb{Q}_p$ satisfying $ \sum_{j=1}^{n}\langle x, τ_j\rangle τ_j =bx$ for all $ x \in \mathbb{Q}^d_p.$ 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 [\textit{IEEE Transactions on Information Theory, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06763 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | p-adic Welch Bounds and p-adic Zauner Conjecture Krishna, K. Mahesh Number Theory Algebraic Geometry 12J25, 46S10, 47S10, 11D88 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 $\{τ_j\}_{j=1}^n$ be a collection in $\mathbb{Q}_p^d$ satisfying (i) $\langle τ_j, τ_j\rangle =1$ for all $1\leq j \leq n$ and (ii) there exists $b \in \mathbb{Q}_p$ satisfying $ \sum_{j=1}^{n}\langle x, τ_j\rangle τ_j =bx$ for all $ x \in \mathbb{Q}^d_p.$ 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 [\textit{IEEE Transactions on Information Theory, 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. |
| title | p-adic Welch Bounds and p-adic Zauner Conjecture |
| topic | Number Theory Algebraic Geometry 12J25, 46S10, 47S10, 11D88 |
| url | https://arxiv.org/abs/2209.06763 |