# Thermodynamics of Quantum Feedback Cooling

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## Abstract

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## 1. Introduction

## 2. Feedback Cooling Algorithm

#### 2.1. Coherent Feedback Control

#### 2.2. Stages of the Feedback Cooling Algorithm

**Figure 1.**Sketch of the four steps of the spin cooling algorithm. The polarization bias of the marginals is illustrated by means of their effective “spin temperatures”, indicated with thermometers, and the correlations and residual coherence are depicted as shaded yellow areas. First, $\mathsf{S}$ and $\mathsf{A}$ are initialized in an uncorrelated state with polarization biases ${\u03f5}_{\mathsf{S}}<{\u03f5}_{\mathsf{A}}$. The measurement unitary ${\widehat{U}}_{\text{m}}$ correlates the two parts, yielding marginals with biases ${\u03f5}_{\mathsf{S}}cos{\phi}^{2}$ and zero, respectively (see the text for details on notation). After the application of the feedback unitary ${\widehat{U}}_{\text{f}}$, most correlations are wiped out as $\mathsf{S}$ is mapped to the more polarized target ${\widehat{\rho}}_{\mathsf{S}}$, with polarization bias ${\u03f5}_{\mathsf{A}}sin\phi $. The marginal of $\mathsf{A}$ is then dissipatively reset to ${\widehat{\rho}}_{0}^{(\mathsf{A})}$.

#### 2.2.1. Initialization

#### 2.2.2. (Pre-)Measurement

#### 2.2.3. Feedback

#### 2.2.4. Reset of the Ancilla

## 3. Thermodynamic Analysis

#### 3.1. Energy Balance

#### 3.2. Performance of Feedback Cooling

**Figure 2.**(

**a**) Coefficient of performance and (

**b**) figure of merit χ versus the entropy reduction on the registers $\mathcal{P}={k}_{B}T\Delta {S}_{0,\text{f}}^{(\mathsf{S})}$ for fixed initial polarization bias ${\u03f5}_{\mathsf{S}}=0.4$ and different measurement directions: $\phi =0$ (solid), $\phi =\pi /4$ (dashed) and $\phi =2\pi /5$ (dotted). In both plots, the bias of the ancillas ${\u03f5}_{\mathsf{A}}$ ranges from ${\u03f5}_{\mathsf{S}}$ to one, and the temperature is $T=1$. The part of the curves falling inside the cooling window $\frac{{\u03f5}_{\mathsf{S}}}{sin\phi}<{\u03f5}_{\mathsf{A}}<1$ is depicted in black, whereas configurations for which $\Delta {E}_{0,\text{f}}^{(\mathsf{S})}<0$ (i.e., no real cooling occurs) lie within the shaded red areas. The grey regions correspond to inaccessible configurations, and the optimal working points $\{{\mathcal{P}}^{\star},{\epsilon}^{\star}\}$ and $\{{\mathcal{P}}^{\star},{\chi}^{\star}\}$ are indicated with dot-dashed blue lines.

## 4. Information-Theoretic Analysis

#### 4.1. Entanglement

**Figure 3.**(

**a**) Entanglement of formation $\mathcal{E}({\widehat{\varrho}}_{\text{m}})$, (

**b**) mutual information $I({\widehat{\varrho}}_{\text{m}})$ and (

**c**) quantum discord ${\delta}_{\mathsf{A}}({\widehat{\varrho}}_{\text{m}})$ evaluated after the measurement step, versus the entropy reduction rate $\mathcal{P}$ and the figure of merit χ. As in Figure 2, the shaded grey areas, the dashed red curve and dot-dashed blue curve correspond to inaccessible configurations, the threshold towards effective cooling and the optimal operation points, respectively. The dotted white line marks configurations above which the feedback unitary ${\widehat{U}}_{\text{f}}$ becomes capable of extracting work from ${\widehat{\rho}}_{\text{m}}$ (cf. Equation (7)). Finally, the dark shaded grey area of (a) corresponds to working points with zero entanglement between $\mathsf{S}$ and $\mathsf{A}$. We have set ${\u03f5}_{\mathsf{S}}=0.4$ and $T=1$.

#### 4.2. Total, Quantum and Classical Correlations

## 5. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## Appendix: Explicit Formula for the Quantum Mutual Information

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Liuzzo-Scorpo, P.; Correa, L.A.; Schmidt, R.; Adesso, G.
Thermodynamics of Quantum Feedback Cooling. *Entropy* **2016**, *18*, 48.
https://doi.org/10.3390/e18020048

**AMA Style**

Liuzzo-Scorpo P, Correa LA, Schmidt R, Adesso G.
Thermodynamics of Quantum Feedback Cooling. *Entropy*. 2016; 18(2):48.
https://doi.org/10.3390/e18020048

**Chicago/Turabian Style**

Liuzzo-Scorpo, Pietro, Luis A. Correa, Rebecca Schmidt, and Gerardo Adesso.
2016. "Thermodynamics of Quantum Feedback Cooling" *Entropy* 18, no. 2: 48.
https://doi.org/10.3390/e18020048