Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture

Let $(\mathrm{\Omega },\mu )$, $(\mathrm{\Delta },\nu )$ be measure spaces and $\{{\tau }_{\alpha }{\}}_{\alpha \in \mathrm{\Omega }}$, $\{{\omega }_{\beta }{\}}_{\beta \in \mathrm{\Delta }}$ be 1-bounded continuous Parseval frames for a Hilbert space $\mathcal{H}$. Then we show that \begin{align} (1) \quad \quad \quad \quad \log (μ(Ω)ν(Δ))\geq S_τ(h)+S_ω(h)\geq -2 \log \left(\frac{1+\displaystyle \sup_{α\in Ω, β\in Δ}|\langleτ_α, ω_β\rangle|}{2}\right) , \quad \forall h \in $\mathcal{H}$_τ\cap $\mathcal{H}$_ω, \end{align} where \begin{align*} &$\mathcal{H}$_τ:= {h_1 \in $\mathcal{H}$: \langle h_1 , τ_α\rangle \neq 0, α\in Ω}, \quad $\mathcal{H}$_ω:= {h_2 \in $\mathcal{H}$: \langle h_2, ω_β\rangle \neq 0, β\in Δ},\ &S_τ(h):= -\displaystyle\int\limits_Ω\left|\left \langle \frac{h}{|h|}, τ_α\right\rangle \right|^2\log \left|\left \langle \frac{h}{|h|}, τ_α\right\rangle \right|^2,dμ(α), \quad \forall h \in $\mathcal{H}$_τ, \ & S_ω(h):= -\displaystyle\int\limits_Δ\left|\left \langle \frac{h}{|h|}, ω_β\right\rangle \right|^2\log \left|\left \langle \frac{h}{|h|}, ω_β\right\rangle \right|^2,dν(β), \quad \forall h \in $\mathcal{H}$_ω. \end{align*} We call Inequality (1) as $\mathbf{\text{Continuous}}\mathbf{\text{Deutsch}}\mathbf{\text{Uncertainty}}\mathbf{\text{Principle}}$. Inequality (1) improves the uncertainty principle obtained by Deutsch $\mathit{\text{[Phys.}}\mathit{\text{Rev.}}\mathit{\text{Lett.,}}\mathit{\text{1983]}}$. We formulate Kraus conjecture for 1-bounded continuous Parseval frames. We also derive continuous Deutsch uncertainty principles for Banach spaces.