# Stochastic Collisional Quantum Thermometry

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## Abstract

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## 1. Introduction

## 2. Quantum Thermometry

#### 2.1. Thermal Fisher Information

#### 2.2. Collisional Thermometry

#### 2.3. Dephasing Interactions

#### 2.4. Role of Correlations

#### 2.5. Parameter Dependence

## 3. Stochastic Approach

#### 3.1. Random Collision Times

#### 3.2. Optimal Measurements

#### 3.3. Partial Swap Interactions

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Plot of the (log of the) ratio between $\Delta $, Equation (12), and ${\mathcal{F}}_{th}$. Positive regions indicate parameter regimes where a thermometic advantage is achievable via the collisional scheme. (

**b**) Mutual information between two adjacent auxiliary units after each has interacted with the system via a $ZZ$ interaction for a deterministic collisional therometry protocol. (

**c**) Measure of the interdependence between $\gamma $ and $\overline{n}$ captured by Equation (15) for a deterministic protocol with $N=2$ (arbitrary choice). In all panels, the area captured by the dashed black line represents the region in parameter space where the scheme achieves an advantage over the thermal QFI. The white line corresponds to the value of $\overline{n}$ where the QFI is maximal.

**Figure 2.**Comparison of the value of the quantum Fisher information for various Weibull distributions of the collision time interval (see Equation (16)), with the deterministic case, for $\overline{n}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}2$. Similar behavior is seen for other values of temperature above $\overline{n}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1.5$. Inset: Distributions for various values of k shown in the main panel.

**Figure 3.**Comparison of the ratio between $\Delta $ for a Weibull distribution for the exponential distribution, i.e., $k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$, and a deterministic equally spaced waiting time distribution. The green plane represents the crossing point where one term becomes larger than the other. $\gamma {\tau}_{SE}$ is the average time between collisions.

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**MDPI and ACS Style**

O’Connor, E.; Vacchini, B.; Campbell, S.
Stochastic Collisional Quantum Thermometry. *Entropy* **2021**, *23*, 1634.
https://doi.org/10.3390/e23121634

**AMA Style**

O’Connor E, Vacchini B, Campbell S.
Stochastic Collisional Quantum Thermometry. *Entropy*. 2021; 23(12):1634.
https://doi.org/10.3390/e23121634

**Chicago/Turabian Style**

O’Connor, Eoin, Bassano Vacchini, and Steve Campbell.
2021. "Stochastic Collisional Quantum Thermometry" *Entropy* 23, no. 12: 1634.
https://doi.org/10.3390/e23121634