# Quantum Thermodynamic Uncertainty Relations, Generalized Current Fluctuations and Nonequilibrium Fluctuation–Dissipation Inequalities

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## Abstract

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## 1. Introduction

#### 1.1. Fluctuations on Center Stage

#### 1.1.1. Nonequilibrium Relations

#### 1.1.2. Fluctuations of Generalized Current

#### 1.2. This Work—Key Findings and Organization

## 2. Stochastic Dynamics of Gaussian Systems: 2nd and Higher-Order Correlations

#### 2.1. Fluctuations and Stochastic Dynamics of Open Quantum Systems

#### 2.2. Fluctuation–Dissipation Inequality and Robertson–Schrödinger Relation

## 3. Energy Flow and Entropy Production

## 4. Quantum Thermodynamic Uncertainty Relation

#### 4.1. Thermal Fluctuation–Dissipation Inequality

#### 4.2. Numerical Results and Quantifying Error

## 5. Nonequilibrium Current, Energy Flow, and Current Fluctuations

#### 5.1. Generalized Current Fluctuations

#### 5.2. Non-Markovianity of the Damping Kernel

## 6. Nonequilibrium Steady State and Connection to Heat Transfer

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Langevin Equation and Uncertainty Relations

#### Appendix A.1. Fluctuation–Dissipation Inequality

#### Appendix A.2. Robertson–Schrödinger Inequality

## Appendix B. Wick’s Theorem for Thermal States

## Appendix C. Fluctuating Energy Flow

## Appendix D. Master Equation, Density Matrix, and Von Neumann Entropy

## Appendix E. Bath Spectral Density and Dissipation Kernel

## Appendix F. Analytical and Numerical Treatment of the Correlation Functions

## Appendix G. Current–Current Fluctuations in Steady-State Heat Transfer

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**Figure 1.**(

**top**) Phase space entropy for positive Wigner functions (Equation (22), normalized to the minimal uncertainty $log[\pi e]$) as a function of time (measured in multiples of the dissipation rate). We employ the bath spectral density of Equation (A48) and use parameters ${\omega}_{0}/{\gamma}_{0}=10$, $\mathsf{\Lambda}/{\omega}_{0}=10$, and dimensions where $\hslash =1$ as well as choose ${\underline{\sigma}}_{0}=\mathrm{diag}[1/2,1/2]$ for the initial conditions. The lower dashed line gives the lower bound prescribed by the fluctuation–dissipation inequality (FDI) at late times (Equations (12), (13) and (22)), which can be connected to quantum fluctuations in the coupled system+bath system. The upper dashed line gives the exact late time limit of the Gaussian evolution (Equations (22) and (A12)), which also includes thermal fluctuations (see Section 4.1). (

**bottom**) Late-time quantum uncertainty [Equations (9) and (A13)] as a function of $\hslash \beta {\omega}_{0}$, i.e., a measure of the respective impact of quantum or thermal fluctuations. For finite system–bath coupling (black, solid line; ${\omega}_{0}/{\gamma}_{0}=1$), the uncertainty always exceeds the minimal bound of $1/4$ given by the Robertson–Schrödinger equation and saturates the fluctuation–dissipation inequality for $\hslash \beta {\omega}_{0}\gg 1$ (gray, horizontal, dashed line). This discrepancy fades for smaller coupling (gray, solid line; ${\omega}_{0}/{\gamma}_{0}=100$). Additionally, for $\hslash \beta {\omega}_{0}\ll 1$, thermal fluctuations start to prevail over the quantum fluctuations, and the more accurate bound (comparing to the FDI) can be provided by the thermal fluctuation–dissipation inequality (TFDI; gray, dashed, nonhorizontal line; see Section 4.1).

**Figure 2.**Numerical evaluation of the symmetric fluctuations ${\langle \dot{\widehat{\mathcal{Q}}}(t)\dot{\widehat{\mathcal{Q}}}({t}^{\prime})\rangle}_{\mathrm{s}}$ of the system’s momentum operator solely connected to the fluctuating dynamics (see Equation (15)) as a function of time in multiples of the dissipation rate ${\gamma}_{0}$. We employ the $\mathsf{\Lambda}$-model in Equation (A48) for the bath spectral density and use parameters ${\omega}_{0}/{\gamma}_{0}=10$, $\mathsf{\Lambda}/{\omega}_{0}=10$, and $\hslash \beta {\omega}_{0}=2$ and work in dimensionless units where $\hslash =1$. (

**top**) Two-time correlation centered at the dissipation time ${\gamma}_{0}^{-1}$ and normalized to its equal-time correlation at $t={\gamma}_{0}^{-1}$. At $t={\gamma}_{0}^{-1}$, the apparent kink is owed to the numerical resolution in time. The curve is smooth. (

**bottom**) Equal-time correlation normalized to its late-time limit (solid). Lower bound prescribed by thermal fluctuation–dissipation inequality (dashed). The difference between the two is given by $\Delta (t)$ (Equation (28)).

**Figure 3.**Outgoing power connected to the fluctuating dynamics of the system ${P}_{\mathrm{out}}^{\mathrm{fluc}}$ (Equations (17) and (31)) as a function of time in multiples of the dissipation rate ${\gamma}_{0}^{-1}$. Parameters are chosen as in Figure 2. We normalize to the expression for the ingoing power at late times ${P}_{\mathrm{in}}(\infty )$ [Equation (14a)] in order to indicate the balancing of the two at equilibrium (solid line). We further report the corresponding expression using the thermal fluctuation–dissipation inequality, i.e., replacing $\nu \to (2/\beta )\gamma $ in the numerical evaluation, which is always smaller than the full expression (dashed). Lastly, we give the upper estimate of Equation (42) (dotted).

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**MDPI and ACS Style**

Reiche, D.; Hsiang, J.-T.; Hu, B.-L.
Quantum Thermodynamic Uncertainty Relations, Generalized Current Fluctuations and Nonequilibrium Fluctuation–Dissipation Inequalities. *Entropy* **2022**, *24*, 1016.
https://doi.org/10.3390/e24081016

**AMA Style**

Reiche D, Hsiang J-T, Hu B-L.
Quantum Thermodynamic Uncertainty Relations, Generalized Current Fluctuations and Nonequilibrium Fluctuation–Dissipation Inequalities. *Entropy*. 2022; 24(8):1016.
https://doi.org/10.3390/e24081016

**Chicago/Turabian Style**

Reiche, Daniel, Jen-Tsung Hsiang, and Bei-Lok Hu.
2022. "Quantum Thermodynamic Uncertainty Relations, Generalized Current Fluctuations and Nonequilibrium Fluctuation–Dissipation Inequalities" *Entropy* 24, no. 8: 1016.
https://doi.org/10.3390/e24081016