# Quantum Euler Relation for Local Measurements

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## Abstract

**:**

## 1. Introduction

## 2. The Information Gain

#### 2.1. Positive Operator-Valued Measures

#### 2.2. Maximal Information Gain

#### Illustrative Example:

#### 2.3. The Holevo Bound

#### Illustrative Example:

## 3. Projective Local Measurements

#### 3.1. Quantum and Classical Contributions

#### 3.2. Maximum Extractable Work

## 4. Collective Dissipation

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch illustrating fundamentally different ways one can carry out quantum measurements: either (

**i**) globally or (

**ii**) locally. For global measurements, we are usually interested in quantities such as the information gain ${\mathcal{I}}_{g}$ and the Holevo bound ${\chi}_{\mathcal{S}}$. For local measurements, the direct quantities we measure are the local counterparts of ${\mathcal{I}}_{g}$ and ${\chi}_{\mathcal{S}}$: (${\mathcal{I}}_{g}^{A}$, ${\mathcal{I}}_{g}^{B}$, ${\chi}_{A}$, ${\chi}_{B}$).

**Figure 2.**Plots of the information gain ${\mathcal{I}}_{g}$ and the right hand side of Equation (27) as a function of c.

**Figure 3.**Plots of the information gain ${\mathcal{I}}_{g}$ and the right hand side of Equation (28) as a function of c.

**Figure 4.**Table summarizing the results of the present manuscript, with the main result shown in red.

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Touil, A.; Weber, K.; Deffner, S.
Quantum Euler Relation for Local Measurements. *Entropy* **2021**, *23*, 889.
https://doi.org/10.3390/e23070889

**AMA Style**

Touil A, Weber K, Deffner S.
Quantum Euler Relation for Local Measurements. *Entropy*. 2021; 23(7):889.
https://doi.org/10.3390/e23070889

**Chicago/Turabian Style**

Touil, Akram, Kevin Weber, and Sebastian Deffner.
2021. "Quantum Euler Relation for Local Measurements" *Entropy* 23, no. 7: 889.
https://doi.org/10.3390/e23070889