Self--averaging of random quantum dynamics

TL;DR

This paper demonstrates that quantum systems driven by multiple independent random quenches tend to a deterministic evolution as the number of quenches increases, with variance decreasing at least as 1/N, indicating self-averaging behavior.

Contribution

The study provides a rigorous analysis of self-averaging in random quantum dynamics, including asymptotic variance reduction and bounds on the distance between averaged and effective Hamiltonian evolutions.

Findings

Variance of the unitary evolution decreases as 1/N

Distance between averaged and effective evolution scales as 1/N for commuting protocols

Numerical simulations support the theoretical results for non-commuting protocols

Abstract

Stochastic dynamics of a quantum system driven by $N$ statistically independent random sudden quenches in a fixed time interval is studied. We reveal that with growing $N$ the system approaches a deterministic limit indicating self-averaging with respect to its temporal unitary evolution. This phenomenon is quantified by the variance of the unitary matrix governing the time evolution of a finite dimensional quantum system which according to an asymptotic analysis decreases at least as $1/N$. For a special class of protocols (when the averaged Hamiltonian commutes at different times), we prove that for finite $N$ the distance (according to the Frobenius norm) between the averaged unitary evolution operator generated by the Hamiltonian $H$ and the unitary evolution operator generated by the averaged Hamiltonian $\langle H \rangle$ scales as $1/N$. Numerical simulations enlarge this result to a broader class of the non-commuting protocols.