论文标题
非Archimedean Welch边界和非Archimedean Zauner猜想
Non-Archimedean Welch Bounds and Non-Archimedean Zauner Conjecture
论文作者
论文摘要
令$ \ mathbb {k} $为一个非架构（完整的）估值字段满足\ begin {align*} \ left | \ sum_ {j = 1}^{n}^{n}λ_j^2 \ right | = \ right | = \ max_ {1 \ leq j \ leq j \ leq j \ leq j \ leq n} \ mathbb {k}，1 \ leq j \ leq n，\ forall n \ in \ mathbb {n}。 \ end {align*}对于$ d \ in \ mathbb {n} $，令$ \ mathbb {k}^d $为标准$ d $ - d $ -dimemensional non-Archimedean Hilbert Space。令$ m \ in \ mathbb {n} $和$ \ text {sym}^m（\ mathbb {k}^d）$是对称m-Tensors的非Archimedean Hilbert Space。我们证明了以下结果。如果$ \ {τ_j\} _ {j = 1}^n $是$ \ mathbb {k}^d $满足$ \ langleτ_j，τ_j\ rangle = 1 $的集合x \ mapsto \ sum_ {j = 1}^n \ langle x，τ_j^{\ otimes m} \ rangleτ_j^{\ otimes m} \ in \ in \ in \ text {sym} {sym}^m（\ sym}^m（\ mathb {k}^d） \ quad \ max_ {1 \ leq j，k \ leq n，j \ neq k} \ {| n |，|，| \ langleτ_j，τ_k\ w \ rangle |^{2m} {2m} \} \} \} \} \ geq Q Q Q Q Q \ freac \ frac {| n |^2} |^2} |^|^2} {| weft | weft | weft | weft | weft | weft | weft | weft | weft | w | w | }。 \ end {align}我们将不等式（1）称为Welch [\ textit {IEEE Transactions thronge of Information Exewement oon Information Exewhere，1974}]获得的韦尔奇界限的非架构版本。我们制定了非Archimedean Zauner猜想。
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 $\{τ_j\}_{j=1}^n$ is a collection in $\mathbb{K}^d$ satisfying $\langle τ_j, τ_j\rangle =1$ for all $1\leq j \leq n$ and the operator $\text{Sym}^m(\mathbb{K}^d)\ni x \mapsto \sum_{j=1}^n\langle x, τ_j^{\otimes m}\rangle τ_j^{\otimes m} \in \text{Sym}^m(\mathbb{K}^d)$ 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 [\textit{IEEE Transactions on Information Theory, 1974}]. We formulate non-Archimedean Zauner conjecture.
