Uncertainty principle in Entropy terms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T18:20:08Z http://mathoverflow.net/feeds/question/120509 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120509/uncertainty-principle-in-entropy-terms Uncertainty principle in Entropy terms Yauhen Radyna 2013-02-01T10:33:53Z 2013-02-01T10:33:53Z <p><strong>Math Questions:</strong></p> <p>Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm</p> <p>$||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2},$</p> <p>and Fourier transform</p> <p>$(F\psi)(\xi) = \int_{\mathbb{R}}{ \psi(t) e^{-2\pi i \xi t} dt },$ which is an isometry in $L_2(\mathbb{R})$.</p> <p>For each $\psi\in L_2(\mathbb{R})$, $||\psi||=1$, consider related probability density function </p> <p>$\rho_\psi = |\psi|^2 \in L_1(\mathbb{R})$, </p> <p>and its differential entropy</p> <p>$H(\psi) = -\int_{\mathbb{R}}{ \rho_\psi(t) \log\rho_\psi(t) dt }.$</p> <blockquote> <p>1) Does the following inequality holds for some constant $C\in\mathbb{R}$?</p> <p>$H(\psi) + H(F\psi) \ge C.$</p> <p>2) What modifications of this inequality relating Fourier transform and some kind of entropy are known?</p> </blockquote> <p><strong>Background:</strong></p> <p>The uncertainty principle of Classical Quantum Mechanics is formulated in terms of variances</p> <p>$\Delta x \cdot \Delta p \ge \hbar/2.$</p> <p>Here $x$ is a result of coordinate measurement (random variable), and $p$ is a result of momentum measurement (another random variable in the same physical experiment), $\Delta$ stands for variance, $\hbar$ is the reduced Planck constant.</p> <p>Coordinate and momentum observables are known to be related by Fourier transform.</p> <p>Besides variance, uncertainty can be measured in terms of entropy. Variance and entropy can be sometimes related, as Rao -- Cramer inequality estimating variance with the inverse of Fisher information hints.</p>