The growth of operator entropy in operator growth

Abstract

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 |∂_tS_K| ≤ 2b₁∆S_K, where b₁ is the first Lanczos coefficient and ∆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.