The growth of operator entropy in operator growth

We study upper bounds on the growth of operator entropy SK in operator growth. Using uncertainty relation, we first prove a dispersion bound on the growth rate |∂tSK| ≤ 2b1∆SK, where b1 is the first Lanczos coefficient and ∆SK is the variance of SK. 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.