Continuous Deutsch Uncertainty Principle and Continuous Kraus ConjectureLet $(Ω, μ)$, $(Δ, ν)$ be measure spaces and $\{τ_α\}_{α\in Ω}$, $\{ω_β\}_{β\in Δ}$ 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 \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.
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