The growth of operator entropy in operator growth

TL;DR

This paper establishes bounds on the growth of operator entropy in quantum systems, linking it to Krylov complexity and providing insights into the long-term behavior in chaotic and integrable models.

Contribution

It introduces a dispersion bound on operator entropy growth and a tighter long-time bound using a universal relation with Krylov complexity.

Findings

The dispersion bound relates entropy growth rate to its variance.

A universal logarithmic relation between Krylov complexity and operator entropy is identified.

The bounds accurately describe entropy growth in chaotic and integrable systems.

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 $|\partial_t S_K|\leq 2b_1 \Delta S_K$, 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.