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- W4387355406 abstract "Let $(Omega, mu)$, $(Delta, nu)$ be measure spaces and ${tau_alpha}_{alphain Omega}$, ${omega_beta}_{beta in Delta}$ 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}|langletau_alpha , omega_betarangle|}{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):= -displaystyleintlimits_{Omega}left|left langle frac{h}{|h|}, tau_alpharightrangle right|^2log left|left langle frac{h}{|h|}, tau_alpharightrangle right|^2,dmu(alpha), quad forall h in mathcal{H}_tau, & S_omega (h):= -displaystyleintlimits_{Delta}left|left langle frac{h}{|h|}, omega_betarightrangle right|^2log left|left langle frac{h}{|h|}, omega_betarightrangle right|^2,dnu(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." @default.
- W4387355406 created "2023-10-05" @default.
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- W4387355406 date "2023-10-02" @default.
- W4387355406 modified "2023-10-06" @default.
- W4387355406 title "Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture" @default.
- W4387355406 doi "https://doi.org/10.48550/arxiv.2310.01450" @default.
- W4387355406 hasPublicationYear "2023" @default.
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