Version 1
: Received: 26 July 2023 / Approved: 26 July 2023 / Online: 27 July 2023 (05:18:36 CEST)

How to cite:
KRISHNA, K. M. Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture. Preprints2023, 2023071844. https://doi.org/10.20944/preprints202307.1844.v1
KRISHNA, K. M. Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture. Preprints 2023, 2023071844. https://doi.org/10.20944/preprints202307.1844.v1

KRISHNA, K. M. Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture. Preprints2023, 2023071844. https://doi.org/10.20944/preprints202307.1844.v1

APA Style

KRISHNA, K. M. (2023). Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture. Preprints. https://doi.org/10.20944/preprints202307.1844.v1

Chicago/Turabian Style

KRISHNA, K. M. 2023 "Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture" Preprints. https://doi.org/10.20944/preprints202307.1844.v1

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}\label{UE} \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 (\ref{UE}) as \textbf{Continuous Deutsch Uncertainty Principle}. Inequality (\ref{UE}) 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.

Copyright:
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