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

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## Abstract

**:**

**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.

## 1. Introduction

## 2. Preliminaries

## 3. Direct-Sum Majorization Relations for General POVM

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Corollary**

**1.**

**Proof**

**of**

**Corollary**

**1.**

**Corollary**

**2.**

**Proof**

**of**

**Corollary**

**2.**

## 4. Comparison of Bounds

#### 4.1. Qubit Observables

#### 4.2. High-Dimensional System

## 5. Multiple Measurements

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

UR | Uncertainty Relation |

EUR | Entropic Uncertainty Relation |

POVM | Positive-Operator-Valued Measure |

PVM | Projection-Valued Measure |

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**Figure 1.**Bounds for the sum of two Shannon entropies rescaled to the logarithm with base 2. (

**a**,

**b**) Plots of the bounds versus the angle $\theta $ at fixed unsharpness parameters (

**a**) $\mu =1$ and (

**b**) $\mu =0.8$; (

**c**,

**d**) Plots of the bounds versus the unsharpness parameter $\mu $ at fixed angles (

**c**) $\theta =\pi /2$ and (

**d**) $\theta =\pi /3$. (Blue solid curves: our direct-sum majorization bound ${\mathcal{B}}_{d2}$ in Equation (22); red dashed curves: Maassen–Uffink bound ${\mathcal{B}}_{MU}$ in Equation (16); orange dotted curves: previous direct-sum majorization bound ${\mathcal{B}}_{d1}$ in Equation (13); and purple dot-dashed curves: tensor-product majorization bound ${\mathcal{B}}_{t}$ in Equation (8)).

**Figure 2.**Plot of ${\mathcal{B}}_{d2}$ (red), ${\mathcal{B}}_{t}$ (blue), and their difference ${\mathcal{B}}_{d2}-{\mathcal{B}}_{t}$ (orange) versus the averaged device uncertainty $\frac{1}{2}(D(F)+D(G))$. The logarithm is taken with respect to the base e, where the most trivial measurement case, i.e., ${\widehat{F}}_{i}={\widehat{G}}_{j}=\widehat{I}/3$ for all i, j, coincides with the point $\frac{1}{2}(D(F)+D(G))=ln3\sim 1.1$.

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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.
https://doi.org/10.3390/e21030270

**AMA Style**

Baek K, Nha H, Son W.
Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements. *Entropy*. 2019; 21(3):270.
https://doi.org/10.3390/e21030270

**Chicago/Turabian Style**

Baek, Kyunghyun, Hyunchul Nha, and Wonmin Son.
2019. "Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements" *Entropy* 21, no. 3: 270.
https://doi.org/10.3390/e21030270