# Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements

## TL;DR

This paper develops improved entropic uncertainty relations for generalized quantum measurements using a novel direct-sum majorization approach, extending previous bounds and applicable to multiple observables.

## Contribution

It introduces a new direct-sum majorization relation for POVMs that significantly enhances entropic uncertainty bounds compared to prior methods.

## Key findings

- The new bound outperforms previous majorization-based bounds.
- The approach is effective for qubit and higher-dimensional unsharp observables.
- It extends to multiple measurements, broadening applicability.

## Abstract

We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches [S. Friendland, V. Gheorghiu and G. Gour, Phys. Rev. Lett. 111, 230401 (2013); A. E. Rastegin and K. \.Zyczkowski, J. Phys. A, 49, 355301 (2016)], particularly by extending the direct-sum majorization relation first introduced in [\L. Rudnicki, Z. Pucha{\l}a and K. \.{Z}yczkowski, Phys. Rev. A 89, 052115 (2014)]. We illustrate the usefulness of our uncertainty relations by considering a pair of qubit observables in a two-dimensional system and randomly chosen unsharp observables in a three-dimensional system. We also demonstrate that our bound tends to be stronger than the generalized Maassen--Uffink bound with an increase in the unsharpness effect. Furthermore, we extend our approach to the case of multiple POVM measurements, thus making it possible to establish entropic uncertainty relations involving more than two observables.