# A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

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*Quantum Reports*in 2024–2025)

## Abstract

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## 1. Introduction

## 2. Preliminaries on Unravelings and Trajectories

## 3. The Model

## 4. The $\mathit{n}=\mathbf{2}$ Case

## 5. Dynamics of the Spectrum

#### 5.1. Stochastic Evolution of the Eigenvalues

#### 5.2. Mapping to Unconstrained Variables

## 6. Stationary State at $\mathit{x}>\mathbf{0}$ and $\mathit{n}\to \infty $

#### 6.1. Finite Time Dynamics

## 7. The Perfect Measurement Dynamics

#### 7.1. Exact Solution at Finite Time

#### 7.2. Relation between the Two Averages

## 8. Exact Results for the Unbiased Ensemble

#### 8.1. Average of Schur’s Polynomials

#### 8.2. Power-Law Symmetric Polynomials

#### 8.3. Calculation of the Moments

#### 8.4. Equivalent Formulations

#### 8.5. Coulomb Gas Regime $\Gamma t\sim O\left(1\right)$

#### 8.6. Universal Regime $\Gamma t=O\left(n\right)$

#### 8.6.1. Scaling of the Edge

#### 8.6.2. Asymptotics at Large $\gamma t$

## 9. Entanglement Entropies for Continuous Monitoring

#### 9.1. Short Time Regime

#### 9.2. Universal Regime

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Identities

## Appendix B. Finite-Time Von Neumann Entropy for the n = 2 Case

## Appendix C. Inverse-Wishart Ensemble

## Appendix D. Equivalent Dyson Brownian Motion

## Appendix E. Kernel

## Appendix F. Large Time Moments from a Saddle Point

## Appendix G. Long-Time Entanglement Entropy

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**Figure 1.**Inverse-Marchenko–Pastur distribution. The spectral density $f\left(\tilde{\lambda}\right)$ for the rescaled eigenvalues $\tilde{\lambda}$ defined in Equation (28) is shown. In the large n limit, it takes the inverse-Marchenko–Pastur form given in Equation (35). In both plots, the orange line displays the theoretical curve, whereas the blue histogram bars are computed after a numerical simulation (with $n=50$) of the weak measurement protocol. (

**a**) The spectral density at $x=0.2$ in the range $[0,4/x]$. At small x, most eigenvalues are located in vicinity of ${\tilde{\lambda}}_{-}\to 1/4$, as an effect of purification. (

**b**) The spectral density at $x=5.0$ in its domain $[{\tilde{\lambda}}_{-},{\tilde{\lambda}}_{+}]$. For larger values of x, the rescaled eigenvalues take finite values around their average $\langle \tilde{\lambda}\rangle =2/x$.

**Figure 2.**Short time behavior. The density ${f}_{\tau}\left(w\right)$ in the short time $t=\tau /4\Gamma \sim 1/\Gamma $ regime, with $\tau $ finite. ${f}_{\tau}\left(w\right)$ features a crossover between a semi-circle at small $\tau $ and a square distribution at larger $\tau $. (

**a**) The small-$\tau $ semi-circle distribution is displayed for $\tau =5$ and increasing n from $n=10$ to $n=100$. The red solid line shows the theoretical $n\to \infty $ curve. (

**b**) The large-$\tau $ square distribution is shown for $\tau =50$ and increasing n, with the red solid line showing the $n\to \infty $ curve. (

**c**) The crossover from semi-circle towards square distribution is shown for increasing $\tau $ from $\tau =5$ to $\tau =50$. All solid lines represent the theoretical density ${w}_{e}w\xb7{f}_{\tau}\left(w\right)$ on the rescaled $w/{w}_{e}$ axis, where ${w}_{e}$ is the edge coordinate for ${f}_{\tau}\left(w\right)$ in the $n\to \infty $ limit.

**Figure 3.**Long-time ranked diffusion. Long-time dynamics of the averaged eigenvalues $\lambda \left(t\right)$ for $x=0$, $\gamma =1$, $n=10$, whereas the maximum eigenvalue tends to one as $\langle {\lambda}_{10}\left(t\right)\rangle \sim 1-{e}^{-4t}$ as time increases, the first $n-1$ eigenvalues are equispaced in logarithmic scale, i.e., $\langle {\lambda}_{\alpha}\left(t\right)\rangle /\langle {\lambda}_{\beta}\left(t\right)\rangle ={e}^{4(\alpha -\beta )t}$.

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**MDPI and ACS Style**

Gerbino, F.; Le Doussal, P.; Giachetti, G.; De Luca, A.
A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems. *Quantum Rep.* **2024**, *6*, 200-230.
https://doi.org/10.3390/quantum6020016

**AMA Style**

Gerbino F, Le Doussal P, Giachetti G, De Luca A.
A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems. *Quantum Reports*. 2024; 6(2):200-230.
https://doi.org/10.3390/quantum6020016

**Chicago/Turabian Style**

Gerbino, Federico, Pierre Le Doussal, Guido Giachetti, and Andrea De Luca.
2024. "A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems" *Quantum Reports* 6, no. 2: 200-230.
https://doi.org/10.3390/quantum6020016