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Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and$\{\tau_\alpha\}_{\alpha\in \Omega}$, $\{\omega_\beta\}_{\beta \in \Delta}$ be1-bounded continuous Parseval frames for a Hilbert space $\mathcal{H}$. Then weshow 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{ContinuousDeutsch Uncertainty Principle}. Inequality (1) improves the uncertaintyprinciple obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We formulateKraus conjecture for 1-bounded continuous Parseval frames. We also derivecontinuous Deutsch uncertainty principles for Banach spaces.

Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture