Universal relation for operator complexity

Abstract

We study Krylov complexity ${C}_{K}$ and operator entropy ${S}_{K}$ in operator growth. We find that for a variety of systems, including chaotic ones and integrable theories, the two quantities always enjoy a logarithmic relation ${S}_{K}\ensuremath{\sim}\text{ln}{C}_{K}$ at long times, where dissipative behavior emerges in unitary evolution. Otherwise, the relation does not hold any longer. Universality of the relation is deeply connected to irreversibility of operator growth.