# Rényi Divergences, Bures Geometry and Quantum Statistical Thermodynamics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Space of Density Operators

#### 2.1. The Bures Geometry

#### 2.2. The Rényi α-Divergences

## 3. Results

#### 3.1. Fundamental Relations

#### 3.2. Geometry, Entropy, and the Thermodynamical Free Energy

#### 3.3. Work and Distance

#### 3.4. Examples

## 4. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Free energy changes for (

**a**) quantum harmonic oscillator with $N=100$ and (

**b**) an ensemble of ${n}_{s}=25$ ($N={2}^{{n}_{s}}$) spin-$1/2$ particles with respect to the scaled temperature $T/\omega $. Black-solid line represents the exact results obtained via calculating $-\Delta \Omega ={T}_{f}ln{Z}_{f}-{T}_{i}ln{Z}_{i}$, while blue-circles and red-stars represents the data obtained from Equations (19) and (41) with $g=\pi $, respectively. All the other parameters are as explained in the text.

**Figure 2.**(

**a**) Extracted work from Rabi system as a quantum Otto engine for ${T}_{2}=0.2$ (black-solid) and ${T}_{2}=0.25$ (blue-dashed); (

**b**) The change in the parameter $\kappa =({\Delta}_{T}{S}_{R}-W)/({T}_{1}lnN)$ for ${T}_{2}=0.2$ (black-solid) and ${T}_{2}=0.25$ (blue-dashed). Corresponding upper bounds ${\zeta}_{1}=3$ and ${\zeta}_{2}=4$ are flagged with red-dotted and red dot–dot–dashed lines. The x-axis is the scaled interaction strength $g/\omega $ in both figures. All the other parameters are as explained in the text.

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Hardal, A.Ü.C.; Müstecaplıoğlu, Ö.E.
Rényi Divergences, Bures Geometry and Quantum Statistical Thermodynamics. *Entropy* **2016**, *18*, 455.
https://doi.org/10.3390/e18120455

**AMA Style**

Hardal AÜC, Müstecaplıoğlu ÖE.
Rényi Divergences, Bures Geometry and Quantum Statistical Thermodynamics. *Entropy*. 2016; 18(12):455.
https://doi.org/10.3390/e18120455

**Chicago/Turabian Style**

Hardal, Ali Ümit Cemal, and Özgür Esat Müstecaplıoğlu.
2016. "Rényi Divergences, Bures Geometry and Quantum Statistical Thermodynamics" *Entropy* 18, no. 12: 455.
https://doi.org/10.3390/e18120455