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I found the inequality in wikipedia https://en.wikipedia.org/wiki/Entropic_uncertainty

$$ H(\phi )\leq \log {\sqrt {2\pi eV(\phi )}}, $$ with $\phi$ as "any probability density function on the real line".

Can anyone point to the proof of this statement? What about discrete case?

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  • $\begingroup$ The proof is standard and available in wikipedia. In the discrete case, the entropy is upper bounded by $\log M$, where $M$ is the number of possible values of the random variable. $\endgroup$ – Stelios Mar 27 at 12:47

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