# Stochastic Operator Variance: an observable to diagnose noise and scrambling

## TL;DR

This paper introduces the stochastic operator variance (SOV), a new observable to diagnose noise and scrambling in quantum systems, linking it to out-of-time-order correlators and demonstrating its behavior in a stochastic Lipkin-Meshkov-Glick model.

## Contribution

The paper presents the SOV as a novel observable that characterizes noise effects and quantum scrambling, with analytical and numerical analysis in a stochastic model.

## Key findings

- SOV obeys an uncertainty relation.
- SOV dynamics are linked to quantum Lyapunov exponent.
- SOV behavior is demonstrated in a stochastic LMG model.

## Abstract

Noise is ubiquitous in nature, so it is essential to characterize its effects. Considering a fluctuating Hamiltonian, we introduce an observable, the stochastic operator variance (SOV), which measures the spread of different stochastic trajectories in the space of operators. The SOV obeys an uncertainty relation and allows finding the initial state that minimizes the spread of these trajectories. We show that the dynamics of the SOV is intimately linked to that of out-of-time-order correlators (OTOCs), which define the quantum Lyapunov exponent $\lambda$. Our findings are illustrated analytically and numerically in a stochastic Lipkin-Meshkov-Glick (sLMG) Hamiltonian undergoing energy dephasing.