Sketch of Proof
The proof of this tight inequality depends on the so-called (q, p)-norm of the Fourier transformation. (Establishing this norm is the most difficult part of the proof.) From this norm we are able to establish a lower bound on the sum of the (differential) Rényi entropies where For simplicity, we consider this inequality only in one dimension; the extension to multiple dimensions is straightforward and can be found in the literature cited.
Read more about this topic: Entropic Uncertainty
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