We study Krylov complexity CK and operator entropy SK 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 SK~lnCK 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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Zhong-Ying Fan (2022) studied this question.
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