Next Article in Journal
Evaluating the Transient Energy Dissipation in a Centrifugal Impeller under Rotor-Stator Interaction
Previous Article in Journal
On the Diversity-Based Weighting Method for Risk Assessment and Decision-Making about Natural Hazards
Previous Article in Special Issue
Limiting Uncertainty Relations in Laser-Based Measurements of Position and Velocity Due to Quantum Shot Noise
Article Menu
Issue 3 (March) cover image

Export Article

Open AccessArticle

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

Department of Physics, Texas A&M University at Qatar, Education City, P.O. Box 23874 Doha, Qatar
Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea
Department of Physics, Sogang University, Mapo-gu, Shinsu-dong, Seoul 121-742, Korea
Authors to whom correspondence should be addressed.
Entropy 2019, 21(3), 270;
Received: 13 December 2018 / Revised: 6 March 2019 / Accepted: 7 March 2019 / Published: 11 March 2019
(This article belongs to the Special Issue Entropic Uncertainty Relations and Their Applications)
PDF [1165 KB, uploaded 11 March 2019]


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 (Friendland, S.; Gheorghiu, V.; Gour, G. Phys. Rev. Lett. 2013, 111, 230401; Rastegin, A.E.; Życzkowski, K. J. Phys. A, 2016, 49, 355301), particularly by extending the direct-sum majorization relation first introduced in (Rudnicki, Ł.; Puchała, Z.; Życzkowski, K. Phys. Rev. A 2014, 89, 052115). 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. View Full-Text
Keywords: entropic uncertainty relations; direct-sum majorization relation; positive-operator-valued measure entropic uncertainty relations; direct-sum majorization relation; positive-operator-valued measure

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

Share & Cite This Article

MDPI and ACS Style

Baek, K.; Nha, H.; Son, W. Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements. Entropy 2019, 21, 270.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top