  We study upper bounds on the growth of operator entropy $S_K$ in operator
growth. Using uncertainty relation, we first prove a dispersion bound on the
growth rate $|\partial_t S_K|\leq 2b_1 \Delta S_K$, where $b_1$ is the first
Lanczos coefficient and $\Delta S_K$ is the variance of $S_K$. However, for
irreversible process, this bound generally turns out to be too loose at long
times. We further find a tighter bound in the long time limit using a universal
logarithmic relation between Krylov complexity and operator entropy. The new
bound describes the long time behavior of operator entropy very well for
physically interesting cases, such as chaotic systems and integrable models.
