Stochastic Operator Variance: an observable to diagnose noise and scrambling

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.