论文标题
模块化韦尔奇界限与应用
Modular Welch Bounds with Applications
论文作者
论文摘要
我们证明了以下两个结果。 \开始{枚举} \ item ling $ \ mathcal {a} $为UNITAL Commutative c*-algebra，$ \ Mathcal {a}^d $是标准Hilbert C*-Module，$ \ Mathcal {A a} $。令$ n \ geq d $。如果$ \ {τ_j\} _ {j = 1}^n $是$ \ Mathcal {a}^d $中的任何载体的收集 \ begin {align*} \ max _ {1 \ leq j，k \ leq n，j \ neq k} \ | \ langleτ_j，τ_k\ rangle ||^{2m} \ geq \ geq \ frac {1} {n-1} {n-1} {n-1} {n-1} {n} \ forall m \ in \ mathbb {n}。 \ end {align*} \ item $ \ mathcal {a} $为$σ$ -finite的交换w*-algebra或交换性AW*-Algebra和$ \ Mathcal {e} $是等级D Hilbert c*-module $ \ nalcal {a} $。令$ n \ geq d $。如果$ / \ begin {align*} \ max _ {1 \ leq j，k \ leq n，j \ neq k} \ | \ langleτ_j，τ_k\ rangle ||^{2m} \ geq \ geq \ frac {1} {n-1} {n-1} {n-1} {n-1} {n} \ forall m \ in \ mathbb {n}。 \ end {align*} \ end {enumerate}结果（1）和（2）将48年前获得的韦尔奇[\ textit {ieee textit {ieee textit {ieee thectit {ieee theckit {ieee {ieee transactions oon Information Thracsions}]的著名结果。我们介绍了模块化框架电势，模块化等距离框架和模块化草个框架的概念。我们为希尔伯特C*模型制定了Zauner的猜想。
We prove the following two results. \begin{enumerate} \item Let $\mathcal{A}$ be a unital commutative C*-algebra and $\mathcal{A}^d$ be the standard Hilbert C*-module over $\mathcal{A}$. Let $n\geq d$. If $\{τ_j\}_{j=1}^n$ is any collection of vectors in $\mathcal{A}^d$ such that $\langle τ_j, τ_j \rangle =1$, $\forall 1\leq j \leq n$, then \begin{align*} \max _{1\leq j,k \leq n, j\neq k}\|\langle τ_j, τ_k\rangle ||^{2m}\geq \frac{1}{n-1}\left[\frac{n}{d+m-1\choose m}-1\right], \quad \forall m \in \mathbb{N}. \end{align*} \item Let $\mathcal{A}$ be a $σ$-finite commutative W*-algebra or a commutative AW*-algebra and $\mathcal{E}$ be a rank d Hilbert C*-module over $\mathcal{A}$. Let $n\geq d$. If $\{τ_j\}_{j=1}^n$ is any collection of vectors in $\mathcal{E}$ such that $\langle τ_j, τ_j \rangle =1$, $\forall 1\leq j \leq n$, then \begin{align*} \max _{1\leq j,k \leq n, j\neq k}\|\langle τ_j, τ_k\rangle ||^{2m}\geq \frac{1}{n-1}\left[\frac{n}{d+m-1\choose m}-1\right], \quad \forall m \in \mathbb{N}. \end{align*} \end{enumerate} Results (1) and (2) reduce to the famous result of Welch [\textit{IEEE Transactions on Information Theory, 1974}] obtained 48 years ago. We introduce the notions of modular frame potential, modular equiangular frames and modular Grassmannian frames. We formulate Zauner's conjecture for Hilbert C*-modules.
