Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture (2310.01450v1)

Abstract: Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and ${\tau_\alpha}{\alpha\in \Omega}$, ${\omega\beta}{\beta \in \Delta}$ be 1-bounded continuous Parseval frames for a Hilbert space $\mathcal{H}$. Then we show that \begin{align} (1) \quad \quad \quad \quad \log (\mu(\Omega)\nu(\Delta))\geq S\tau(h)+S_\omega (h)\geq -2 \log \left(\frac{1+\displaystyle \sup_{\alpha \in \Omega, \beta \in \Delta}|\langle\tau_\alpha , \omega_\beta\rangle|}{2}\right) , \quad \forall h \in \mathcal{H}\tau \cap \mathcal{H}\omega, \end{align} where \begin{align*} &\mathcal{H}\tau := {h_1 \in \mathcal{H}: \langle h_1 , \tau\alpha \rangle \neq 0, \alpha \in \Omega}, \quad \mathcal{H}\omega := {h_2 \in \mathcal{H}: \langle h_2, \omega\beta \rangle \neq 0, \beta \in \Delta},\ &S_\tau(h):= -\displaystyle\int\limits_{\Omega}\left|\left \langle \frac{h}{|h|}, \tau_\alpha\right\rangle \right|^2\log \left|\left \langle \frac{h}{|h|}, \tau_\alpha\right\rangle \right|^2\,d\mu(\alpha), \quad \forall h \in \mathcal{H}\tau, \ & S\omega (h):= -\displaystyle\int\limits_{\Delta}\left|\left \langle \frac{h}{|h|}, \omega_\beta\right\rangle \right|^2\log \left|\left \langle \frac{h}{|h|}, \omega_\beta\right\rangle \right|^2\,d\nu(\beta), \quad \forall h \in \mathcal{H}_\omega. \end{align*} We call Inequality (1) as \textbf{Continuous Deutsch Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We formulate Kraus conjecture for 1-bounded continuous Parseval frames. We also derive continuous Deutsch uncertainty principles for Banach spaces.