Functional Deutsch Uncertainty Principle

Abstract: Let ${f_j}{j=1}^n$ and ${g_k}{k=1}^m$ be Parseval p-frames for a finite dimensional Banach space $\mathcal{X}$. Then we show that \begin{align} (1) \quad\quad\quad\quad \log (nm)\geq S_f (x)+S_g (x)\geq -p \log \left(\displaystyle\sup_{y \in \mathcal{X}f\cap \mathcal{X}_g, |y|=1}\left(\max{1\leq j\leq n, 1\leq k\leq m}|f_j(y)g_k(y)|\right)\right), \quad \forall x \in \mathcal{X}f\cap \mathcal{X}_g, \end{align} where \begin{align*} &\mathcal{X}_f:= {z\in \mathcal{X}: f_j(z)\neq 0, 1\leq j \leq n}, \quad \mathcal{X}_g:= {w\in \mathcal{X}: g_k(w)\neq 0, 1\leq k \leq m},\ &S_f (x):= -\sum{j=1}^{n}\left|f_j\left(\frac{x}{|x|}\right)\right|^p\log \left|f_j\left(\frac{x}{|x|}\right)\right|^p, \quad S_g (x):= -\sum_{k=1}^{m}\left|g_k\left(\frac{x}{|x|}\right)\right|^p\log \left|g_k\left(\frac{x}{|x|}\right)\right|^p, \quad \forall x \in \mathcal{X}_g. \end{align*} We call Inequality (1) as \textbf{Functional Deutsch Uncertainty Principle}. For Hilbert spaces, we show that Inequality (1) reduces to the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We also derive a dual of Inequality (1).