# Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources

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## Abstract

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## 1. Introduction

## 2. Quantum Setup and Notation

#### Removing Coherences

## 3. Classical Setup and Notation

#### 3.1. Energy-Shell Inhomogeneities

#### 3.2. Removing Inhomogeneities

## 4. Quantum–Classical Comparison

#### 4.1. Energy Equivalence Classes

#### 4.2. An Unfair Comparison

#### 4.3. A Fair Comparison

## 5. Dropping the Isoenergetic Constraint

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**Schematic illustration of the quantum process described in the text. The system begins in state ${\widehat{\rho}}_{i}$, then evolves in contact with a thermal bath to a final state ${\widehat{\rho}}_{f}$ as the Hamiltonian is driven through a cycle from $\widehat{H}(0)={\widehat{H}}_{0}$ to $\widehat{H}(\tau )={\widehat{H}}_{0}$. We impose the constraint $\mathrm{diag}\phantom{\rule{0.166667em}{0ex}}{\widehat{\rho}}_{i}=\mathrm{diag}\phantom{\rule{0.166667em}{0ex}}{\widehat{\rho}}_{f}$, which indicates that the initial and final energy distributions are identical, while the coherences may differ.

**Figure 2.**Schematic illustration of the classical process. The system begins in state ${\rho}_{i}(\Gamma )$, then evolves in contact with a thermal bath to a final state ${\rho}_{f}(\Gamma )$ as the Hamiltonian is driven through a cycle from $H(\Gamma ,0)={H}_{0}(\Gamma )$ to $H(\Gamma ,\tau )={H}_{0}(\Gamma )$. The constraint $\mathrm{diag}\phantom{\rule{0.166667em}{0ex}}{\rho}_{i}(\Gamma )=\mathrm{diag}\phantom{\rule{0.166667em}{0ex}}{\rho}_{f}(\Gamma )$ indicates that the initial and final energy distributions are identical, while inhomogeneities may differ.

**Figure 3.**An ideal gas inside a box of volume ${L}^{3}$. The value of $\alpha \in (0,1]$ parametrizes a family of energy equivalence classes, with $\alpha =1$ corresponding to thermal class ${\mathsf{\Pi}}^{c}$. See text for details.

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**MDPI and ACS Style**

Smith, A.; Sinha, K.; Jarzynski, C.
Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources. *Entropy* **2022**, *24*, 474.
https://doi.org/10.3390/e24040474

**AMA Style**

Smith A, Sinha K, Jarzynski C.
Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources. *Entropy*. 2022; 24(4):474.
https://doi.org/10.3390/e24040474

**Chicago/Turabian Style**

Smith, Andrew, Kanupriya Sinha, and Christopher Jarzynski.
2022. "Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources" *Entropy* 24, no. 4: 474.
https://doi.org/10.3390/e24040474