# Axiomatic Characterization of the Quantum Relative Entropy and Free Energy

^{*}

## Abstract

**:**

## 1. Introduction

- Continuity: For fixed $\sigma $, the map $\rho \mapsto S\left(\rho \right|\left|\sigma \right)$ is continuous [15].
- Data-processing inequality: For any quantum channel T we have,$$S\left(T\right(\rho \left)\right|\left|T\right(\sigma \left)\right)\le S\left(\rho \right|\left|\sigma \right).$$
- Additivity:$$S({\rho}_{1}\otimes {\rho}_{2}\left|\right|{\sigma}_{1}\otimes {\sigma}_{2})=S({\rho}_{1}\left|\right|{\sigma}_{1})+S({\rho}_{2}\left|\right|{\sigma}_{2}).$$
- Super-additivity: For any bipartite state ${\rho}_{1,2}$ with marginals ${\rho}_{1},{\rho}_{2}$ we have$$S({\rho}_{1,2}\left|\right|{\sigma}_{1}\otimes {\sigma}_{2})\ge S({\rho}_{1}\left|\right|{\sigma}_{1})+S({\rho}_{2}\left|\right|{\sigma}_{2}).$$

## 2. Axiomatic Derivation of Quantum Relative Entropy

**Theorem**

**1.**

**Theorem**

**2.**

**Lemma**

**1.**

- The map $\rho \mapsto f(\rho ,\sigma )$ is continuous for any fixed σ.
- Additivity: $f({\rho}_{1}\otimes {\rho}_{2},{\sigma}_{1}\otimes {\sigma}_{2})={\sum}_{i=1}^{2}f({\rho}_{i},{\sigma}_{i})$.
- Super-additivity:$$f({\rho}_{1,2},{\sigma}_{1}\otimes {\sigma}_{2})\ge f({\rho}_{1}\otimes {\rho}_{2},{\sigma}_{1}\otimes {\sigma}_{2}).$$

**Proof.**

## 3. Uniqueness of the Free Energy

**Lemma**

**2.**

#### 3.1. Catalysts and Correlations

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 3.2. Free Energy as a Unique Measure of Non-Equilibrium

- Continuity: For fixed Hamiltonian H, the map $\rho \mapsto {M}_{\beta}(\rho ,H)$ is continuous.
- Additivity:$${M}_{\beta}({\rho}_{1}\otimes {\rho}_{2},{H}_{1}+{H}_{2})={M}_{\beta}({\rho}_{1},{H}_{1})+{M}_{\beta}({\rho}_{2},{H}_{2}).$$
- Monotonicity:
- (a)
- Monotonicity:${M}_{\beta}(\rho ,H)\ge {M}_{\beta}(\sigma ,K)$ if $G(\rho ,H)=(\sigma ,K)$.
- (b)
- Catalytic monotonicity:${M}_{\beta}(\rho ,H)\ge {M}_{\beta}(\sigma ,K)$ if $(\rho ,H)\stackrel{\mathit{c}}{>}(\sigma ,K)$.
- (c)
- Marginal-catalytic monotonicity:${M}_{\beta}(\rho ,H)\ge {M}_{\beta}(\sigma ,K)$ if $(\rho ,H)\stackrel{mc}{>}(\sigma ,K)$.
- (d)
- Correlated-catalytic monotonicity:${M}_{\beta}(\rho ,H)\ge {M}_{\beta}(\sigma ,K)$ if $(\rho ,H)\stackrel{cc}{>}(\sigma ,K)$.

**Lemma**

**3.**

**Proof.**

**Theorem**

**3.**

**Result**

**1.**

## 4. Connection to Entropy Production in Master Equations

## 5. Discussion and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Gauge Invariance of ${M}_{\beta}$

## Appendix B. Rank-Decreasing Quantum Channels

## Appendix C. Proof of Theorem 3 and Other Equivalences

**Lemma**

**A1.**

- (i)
- ${M}_{\beta}$ fulfills monotonicity III.a ⇔ ${\mathcal{M}}_{\beta}$ fulfills the data-processing inequality (DPI) (31).
- (ii)
- ${M}_{\beta}$ fulfills catalytic monotonicity III.b and additivity II. ⇔ ${\mathcal{M}}_{\beta}$ fulfills additivity (30) and the DPI (31).
- (iii)
- (iv)

**Proof.**

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Wilming, H.; Gallego, R.; Eisert, J.
Axiomatic Characterization of the Quantum Relative Entropy and Free Energy. *Entropy* **2017**, *19*, 241.
https://doi.org/10.3390/e19060241

**AMA Style**

Wilming H, Gallego R, Eisert J.
Axiomatic Characterization of the Quantum Relative Entropy and Free Energy. *Entropy*. 2017; 19(6):241.
https://doi.org/10.3390/e19060241

**Chicago/Turabian Style**

Wilming, Henrik, Rodrigo Gallego, and Jens Eisert.
2017. "Axiomatic Characterization of the Quantum Relative Entropy and Free Energy" *Entropy* 19, no. 6: 241.
https://doi.org/10.3390/e19060241