Optimality of entropic uncertainty relations

TL;DR

This paper investigates the optimality of entropic uncertainty relations, establishing new bounds, disproving a conjecture, and providing insights into the structure of these relations through analytical and numerical methods.

Contribution

It characterizes the optimal lower bounds for entropic uncertainty relations, disproves a previous conjecture, and extends existing results in the field.

Findings

Disproved a conjecture by Englert et al.

Established optimal lower bounds for entropic uncertainty relations.

Presented conjectures based on numerical and analytical analysis.

Abstract

The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be non-optimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work we establish optimal uncertainty relations by characterising the optimal lower bound in scenarios similar to the Maassen-Uffink type. We disprove a conjecture by Englert et al. and generalise various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures.