Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$

If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will fluctuate forever. (stabilization means the fluctuation approaching to zero, but the derivative can still be large, so here, we want to show that it will endlessly oscillate and never be damped.)

Mathematically, $H$ is an Hermitian matrix and $\rho(t)$ is a square matrix with trace one, positiveness and self-adjoint.

We need a mathematical proof that particularly by using the fact that $[H, \rho(0)] \neq 0$, i.e. $H\rho(0)\neq\rho(0)H$