Modular Welch Bounds with Applications

Abstract

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 $\{\tau_j\}_{j=1}^n$ is any collection of vectors in $\mathcal{A}^d$ such that $\langle \tau_j, \tau_j \rangle =1$, $\forall 1\leq j \leq n$, then \begin{align*} \max _{1\leq j,k \leq n, j\neq k}\|\langle \tau_j, \tau_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 $\sigma$-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 $\{\tau_j\}_{j=1}^n$ is any collection of vectors in $\mathcal{E}$ such that $\langle \tau_j, \tau_j \rangle =1$, $\forall 1\leq j \leq n$, then \begin{align*} \max _{1\leq j,k \leq n, j\neq k}\|\langle \tau_j, \tau_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.