Uncertainty principle in Entropy terms

2013-02-01T10:33:53Z

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?

Background:

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.

Uncertainty principle in Entropy terms - MathOverflow

Uncertainty principle in Entropy terms