# Quantum Bounds on the Generalized Lyapunov Exponents

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## Abstract

**:**

## 1. Introduction

## 2. Classical Generalized Lyapunov Exponents

- ${L}_{2q}/2q$ are an increasing function of the order q:$$\frac{d}{dq}\left(\frac{{L}_{2q}}{2q}\right)\ge 0\phantom{\rule{4pt}{0ex}};$$
- The GLEs are always bounded by the linear behavior:$${L}_{2q}\ge 2q{\lambda}_{1}\phantom{\rule{4pt}{0ex}}.$$

## 3. Quantum Generalized Lyapunov Exponents at Infinite Temperature

#### 3.1. Properties of the Infinite Temperature Quantum GLE

- ${L}_{2q}^{\left(0\right)}/2q$ is an increasing function of q;
- The following inequality holds:$${L}_{2q}^{\left(0\right)}\ge 2q{\lambda}_{1}^{\left(0\right)}\phantom{\rule{4pt}{0ex}},$$

#### 3.2. A Semi-Classical Example: The Quantum Kicked Top

## 4. Thermal Quantum Generalized Exponents

#### 4.1. From Commutators to OTOCS

#### 4.2. Product Space, Fluctuation-Dissipation Theorem and Bound

#### 4.2.1. Fluctuation–Dissipation in the Replicated Space

#### 4.2.2. The Bound

#### 4.3. Distribution Functions

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Dynamics of ${G}_{2q}\left(t\right)$ in Equation (3) for different values of $q=1\xf719$ as a function of time for $N=1600$.

**Figure 2.**Generalized Lyapunov exponents fitted from Figure 1. (

**Left**) ${L}_{2q}$ as a function of q, contrasted with the actual Lyapunov exponent ${\lambda}_{1}$ and the maximal expanding rate ${\lambda}_{max}$, obtained by a fit of these data at large q. (

**Right**) ${L}_{2q}/2q$ as a function of the moment q, from which we extract the maximal Lyapunov exponent ${\lambda}_{1}$.

**Figure 3.**Spectrum of the square commutator ${g}_{i}^{2}\left(t\right)$, compared with ${G}_{2}\left(t\right)$ and the exponential growth with ${L}_{2}$, with the maximal expansion rate ${\lambda}_{max}$ and with ${\lambda}_{1}$ for $N=1600$.

**Figure 4.**Large deviation properties of the spectrum of the square-commutator for $N=1600$. (

**Left**) Numerical distributions of ${\lambda}_{t}^{i}=ln\left({g}_{i}\left(t\right)\right)/2t$ with ${g}_{i}^{2}\left(t\right)$ the eigenvalues of Equation (20) at different times $t=3,4,5$. (

**Right**) $-lnP\left(\lambda \right)/t$ with $P\left(\lambda \right)$ the empirical distribution.

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Pappalardi, S.; Kurchan, J.
Quantum Bounds on the Generalized Lyapunov Exponents. *Entropy* **2023**, *25*, 246.
https://doi.org/10.3390/e25020246

**AMA Style**

Pappalardi S, Kurchan J.
Quantum Bounds on the Generalized Lyapunov Exponents. *Entropy*. 2023; 25(2):246.
https://doi.org/10.3390/e25020246

**Chicago/Turabian Style**

Pappalardi, Silvia, and Jorge Kurchan.
2023. "Quantum Bounds on the Generalized Lyapunov Exponents" *Entropy* 25, no. 2: 246.
https://doi.org/10.3390/e25020246