Continuous Welch bounds with Applications

Let $(\Omega, \mu)$ be a measure space and $\{\tau_\alpha\}_{\alpha\in \Omega}$ be a normalized continuous Bessel family for a finite dimensional Hilbert space $\mathcal{H}$ of dimension $d$. If the diagonal $\Delta\coloneqq \{(\alpha, \alpha):\alpha \in \Omega\}$ is measurable in the measure space $\Omega\times \Omega$, then we show that \begin{align*} \sup _{\alpha, \beta \in \Omega, \alpha\neq \beta}|\langle \tau_\alpha, \tau_\beta\rangle |^{2m}\geq \frac{1}{(\mu\times\mu)((\Omega\times\Omega)\setminus\Delta)}\left[\frac{ \mu(\Omega)^2}{{d+m-1 \choose m}}-(\mu\times\mu)(\Delta)\right], \quad \forall m \in \mathbb{N}. \end{align*} This improves 47 years old celebrated result of Welch [\textit{IEEE Transactions on Information Theory, 1974}]. We introduce the notions of continuous cross correlation and frame potential of Bessel family and give applications of continuous Welch bounds to these concepts. We also introduce the notion of continuous Grassmannian frames.