Self-averaging of random quantum dynamics

The 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 increasing 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 evolution unitary operator generated by the Hamiltonian H and the unitary evolution operator generated by the averaged Hamiltonian < H > scales as 1/N. Numerical simulations enlarge this result to a broader class of noncommuting protocols.