论文标题
P-Adic Welch边界和P-Adic Zauner猜想
p-adic Welch Bounds and p-adic Zauner Conjecture
论文作者
论文摘要
令$ p $为素数。对于$ d \ in \ mathbb {n} $，令$ \ mathbb {q} _p^d $为标准$ d $ - 维度p-Adic Hilbert Space。令$ m \ in \ mathbb {n} $和$ \ text {sym}^m（\ mathbb {q} _p^d）$是对称m-Tensors的P-Adic Hilbert空间。我们证明了以下结果。让$ \ {τ_j\} _ {j = 1}^n $为$ \ m arthbb {q} _p^d $满足（i）$ \ langleτ_j，τ_j\ rangle = 1 $ for $ 1 \ leq j \ leq n $（ii）$ b b b b b b b b \ b \ b b \ b \ b \ b b \ b b \ b \ b b \ b \ b \ b b \ b b \ b \ b b \ b b \ b b \ b b \ b \ b \ b \ b \ b \ b.满足$ \ sum_ {j = 1}^{n} \ langle x，τ_j\ rangleτ_j= bx $ for All $ x \ in \ Mathbb {q}^d_p。$ \ begin {align} （1）\ quad \ quad \ quad \ max_ {1 \ leq j，k \ leq n，j \ neq k} \ {| n |，|，| \ langleτ_j，τ_k\τ_k\ rangle | rangle |^{2m} m} \ right | }。 \ end {align}我们将不等式（1）称为韦尔奇[\ textit {ieee transactions for Information thronexons oon Information Throweble of Information Thression，1974}]的p-Adic版本。不平等（1）不同于克里希纳（M.我们制定了P-Adic Zauner猜想。
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.
