# Measurement-Based Quantum Thermal Machines with Feedback Control

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Quantum Maxwell’s Demon

#### 3.1. Discrete One-Qubit Measurement

#### 3.2. Discrete Two-Qubit Combined Measurement

#### 3.3. Continuous One- and Two-Qubit Measurement

## 4. Measurement-Assisted Refrigerator

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Quantum Master Equation

## References

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**Figure 1.**Coupled-qubit-based quantum feedback thermal machine. Qubit ${\mathrm{Q}}_{i}$ is attached to a thermal bath with temperature ${T}_{i}$ and is being monitored by the measurement apparatus ${\mathrm{D}}_{i}$ for $i=1,2$. In the case of Maxwell’s demon, ${T}_{1}={T}_{2}=T$, whereas in the case of the measurement-assisted refrigerator, the two baths have different temperatures. Similarly, the two demons can undergo measurements with varying strengths.

**Figure 2.**Heat extracted $\left(Q\right)$ as a function of qubit–qubit coupling strength (Panel (

**a**)) and measurement strength ${\kappa}_{1}$ (Panel (

**b**) for ${\mathrm{F}}_{1}=+1$ and Panel (

**c**) for ${\mathrm{F}}_{1}=-1$). In Panel (

**a**), the black and red curves give the heat extraction for two values of the measurement strength ${\kappa}_{1}$ and for feedback ${\mathrm{F}}_{1}=+1$ (rotation to positive z-axis). Similarly, the purple and blue curves give the heat extraction for feedback ${\mathrm{F}}_{1}=-1$ (rotation to negative z-axis). In Panels (

**b**,

**c**), we plot the heat extraction as a function of ${\kappa}_{1}$ for ${\mathrm{F}}_{1}=+1$ and ${\mathrm{F}}_{1}=-1$, respectively, taking different coupling strengths between the qubits. We take ${\u03f5}_{1}=0.1{k}_{\mathrm{B}}T$, ${\u03f5}_{2}=2{k}_{\mathrm{B}}T$.

**Figure 3.**Heat extracted Q as a function of ${\Delta}_{z}$ for ${\kappa}_{1}={\kappa}_{2}=0.2$. The feedback is represented as $\mathrm{F}=({\mathrm{F}}_{1},\text{}{\mathrm{F}}_{2})$, where ${\mathrm{F}}_{i}$ is the feedback applied to the qubit ${\mathrm{Q}}_{i}$. Finite heat extraction is obtained only when ${\mathrm{F}}_{2}=-1$ (see the dashed black and solid red curves obtained with feedback $\mathrm{F}=(+1,-1)$ and $\mathrm{F}=(-1,-1)$, respectively). The dotted purple and dashed blue curves obtained with feedback $\mathrm{F}=(+1,+1)$ and $\mathrm{F}=(-1,+1)$ lead to the heating of the baths (see the inset). We take the same parameters as Figure 2.

**Figure 4.**Heat extracted (Q) as a function of ${\Delta}_{z}$ for individual measurement (solid red curve) and combined measurement of two qubits (dashed black curve). As feedback, we applied $\mathrm{F}=(+1,-1)$ in both cases. We take ${\u03f5}_{1}=0.1{k}_{\mathrm{B}}T$, ${\u03f5}_{2}=0.5{k}_{\mathrm{B}}T$, ${\kappa}_{1}={\kappa}_{2}=0.3$.

**Figure 5.**Count distribution of the heat extraction for one-qubit continuous measurement (Panel (

**a**)) and the two-qubit combined continuous measurement (Panel (

**b**)) for $\delta t/\tau =0.01$. The dashed lines indicate the averages of the distributions. The simulation is performed for $n=20$ sequential measurements with feedback application only at the end. The distributions are for $N=$ 20,000 simulations. As feedback, we applied ${\mathrm{F}}_{1}=-1$ in the left panel and $\mathrm{F}=(-1,-1)$ in the right panel. We take the same parameters as Figure 2 for ${\u03f5}_{1},{\u03f5}_{2},{k}_{\mathrm{B}}T$.

**Figure 6.**Average and standard deviation of the heat extraction for one-qubit continuous measurements (

**left**panels) and two-qubit combined continuous measurements (

**right**panel) for $\delta t/\tau =0.01$. The simulation is performed for $n=20$ sequential continuous measurements with feedback application only at the end. The distributions are for $N=$ 20,000 simulations. As feedback, we applied ${\mathrm{F}}_{1}=-1$ in the left panel and $\mathrm{F}=(-1,-1)$ in the right panel. In the inset, we show the variation of the signal- (average heat extracted) to-noise (standard deviation of the extracted heat) ratio as a function of ${\Delta}_{z}$. We take the same parameters as Figure 2 for ${\u03f5}_{1},{\u03f5}_{2},{k}_{\mathrm{B}}T$.

**Figure 7.**Heat extracted Q as a function of ${\Delta}_{z}$ in the individual (solid red curve) and the combined (dashed black curve) cases. The circles and crosses represent the average heat extracted $\langle Q\rangle $, whereas the error bars give the respective fluctuations ${\sigma}_{Q}$. We consider the number of measurements $n=20$ and the number of trajectories $N=5000$. As feedback, we applied $\mathrm{F}=(+1,-1)$ in both cases. We take ${\u03f5}_{1}=0.1{k}_{\mathrm{B}}T$, ${\u03f5}_{2}=0.5{k}_{\mathrm{B}}T$, $\delta t/\tau =0.01$.

**Figure 8.**Refrigeration obtained as a result of measurement and a swap operation in the coupled-qubit system attached to two baths with different temperatures. The black dashed line gives the average heat current, whereas the red and blue curves are the heat current obtained for a single trajectory of measurement. We take ${E}_{1}=5{k}_{\mathrm{B}}T$, ${E}_{2}=2{k}_{\mathrm{B}}T$, ${\mathrm{\Gamma}}_{1}={\mathrm{\Gamma}}_{2}=0.05$, $\Delta =0.2{k}_{\mathrm{B}}T$, ${\mathrm{\Gamma}}_{M}=0.02{k}_{\mathrm{B}}T$, ${T}_{1}=1.1T$, ${T}_{2}=T$, and $\delta t=0.01\hslash /{k}_{\mathrm{B}}T$. ${\mathrm{\Gamma}}_{i}$ parameterizes the coupling strength between the coupled-qubit system and the bath $i=1,2$ (see Appendix A).

**Figure 9.**Heat current flowing out of the cold bath as a function of the rotation angle $\mathrm{\Theta}$ for $\Delta =0.5{k}_{\mathrm{B}}T$ (Panel (

**a**)) and $\Delta ={k}_{\mathrm{B}}T$ (Panel (

**b**)). We take ${T}_{1}=1.1T$, ${T}_{2}=T$, ${E}_{1}=5{k}_{\mathrm{B}}T$, ${E}_{2}=2{k}_{\mathrm{B}}T$, ${\mathrm{\Gamma}}_{1}={\mathrm{\Gamma}}_{2}=0.01$.

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## Share and Cite

**MDPI and ACS Style**

Bhandari, B.; Czupryniak, R.; Erdman, P.A.; Jordan, A.N.
Measurement-Based Quantum Thermal Machines with Feedback Control. *Entropy* **2023**, *25*, 204.
https://doi.org/10.3390/e25020204

**AMA Style**

Bhandari B, Czupryniak R, Erdman PA, Jordan AN.
Measurement-Based Quantum Thermal Machines with Feedback Control. *Entropy*. 2023; 25(2):204.
https://doi.org/10.3390/e25020204

**Chicago/Turabian Style**

Bhandari, Bibek, Robert Czupryniak, Paolo Andrea Erdman, and Andrew N. Jordan.
2023. "Measurement-Based Quantum Thermal Machines with Feedback Control" *Entropy* 25, no. 2: 204.
https://doi.org/10.3390/e25020204