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Entropy 2019, 21(3), 270; https://doi.org/10.3390/e21030270

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

1
Department of Physics, Texas A&M University at Qatar, Education City, P.O. Box 23874 Doha, Qatar
2
Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea
3
Department of Physics, Sogang University, Mapo-gu, Shinsu-dong, Seoul 121-742, Korea
*
Authors to whom correspondence should be addressed.
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)
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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 (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
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Baek, K.; Nha, H.; Son, W. Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements. Entropy 2019, 21, 270.

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