Tight Entropic Uncertainty Relations

Alberto Riccardi; Lorenzo Maccone

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arXiv · 2605.01832

Tight Entropic Uncertainty Relations

Alberto Riccardi·Lorenzo Maccone

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Abstract

Entropic uncertainty relations $H(A)+H(B)\geqslant \gamma$ give a nonzero lower bound $\gamma$ to the sum of the Shannon entropies $H$ of the outcome probabilities of incompatible observables $A$ and $B$. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds $\gamma$ typically are state independent. Here we provide a state-independent lower bound $\gamma_s$ that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter $s\to 2$, becomes asymptotically tight for all $A,B$. The bound can be extended to Renyi entropies. After the submission of this paper, we found out that the main results presented here were already on page 6 of https://quantum-journal.org/papers/q-2018-03-30-59/pdf/

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Alberto Riccardi, Lorenzo Maccone. 2026-05-03. Tight Entropic Uncertainty Relations. https://arxiv.org/abs/2605.01832

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