Tight Entropic Uncertainty Relations

Abstract

Entropic uncertainty relations $H(A)+H(B)⩾{\hat{I}}^3$ give a nonzero lower bound ${\hat{I}}^3$ 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 ${\hat{I}}^3$ typically are state independent. Here we provide a state-independent lower bound ${\hat{I}}_s^3$ 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/

We were scooped. The main results of this paper had appeared previously in a paper by Rene Schwonnek, unbeknownst to us. Rene gracefully pointed it out to us