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Math Questions:

Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm

$ ||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2}, $

and Fourier transform

$ (F\psi)(\xi) = \int_{\mathbb{R}}{ \psi(t) e^{-2\pi i \xi t} dt }, $ which is an isometry in $L_2(\mathbb{R})$.

For each $\psi\in L_2(\mathbb{R})$, $||\psi||=1$, consider related probability density function

$\rho_\psi = |\psi|^2 \in L_1(\mathbb{R})$,

and its differential entropy

$ H(\psi) = -\int_{\mathbb{R}}{ \rho_\psi(t) \log\rho_\psi(t) dt }. $

1) Does the following inequality holds for some constant $C\in\mathbb{R}$?

$ H(\psi) + H(F\psi) \ge C. $

2) What modifications of this inequality relating Fourier transform and some kind of entropy are known?


The uncertainty principle of Classical Quantum Mechanics is formulated in terms of variances

$ \Delta x \cdot \Delta p \ge \hbar/2. $

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.

Coordinate and momentum observables are known to be related by Fourier transform.

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.

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5… – Terry Tao Feb 1 '13 at 18:51
@ Terry Tao. Thanks for the link. I could be more observant ;) In fact, I was interested in the functions $\psi\in L_2(K)$, $K$ being a non-archimedean local field. The notion of a standard deviation is unclear there, but entropy fits well. Now I see the way to handle inequalities like this. Though in non-archimedean case the lower bound is zero, in contrast to the real case. – Yauhen Radyna Feb 2 '13 at 0:50
Also check the formulation of entropic uncertainty and the fourier entropy influence conjucture/theorem – Nikos M. Jun 17 '14 at 16:44
Also note that the differential entropy is not the best way to use entropy for continous distr. An alternative is the Kullback-Leibler distance which is used in information geometry – Nikos M. Jun 17 '14 at 16:48

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