Universal relation for operator complexity

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 $S_K\sim \log{C_K}$ 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.

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