Operator Dynamics beyond Scrambling Time using Krylov Complexity

Dynamical probes in chaotic quantum systems such as the
 out-of-time-ordered-correlators ( or OTOCs) can diagnose growth of local
 operators only up to scrambling times t~ log(S), where S is the system
 entropy. However, independent estimates for randomisation of quantum
 states in the Hilbert space requires the much longer Heisenberg time
 scale tH~exp(S). We study the evolution of operator complexity between
 these two vastly different time scales using the notion of Krylov
 complexity. This notion measures growth of operator complexity in a
 specially designed operator basis called the Krylov-basis, for exponentially
 long time scales.
 Using ETH we argue that Krylov complexity in fast scramblers tends to slow
 down from an exponential to a linear law beyond scrambling time.  We also
 introduce the notion of Krylov entropy to characterise how an initially
 localised operator randomises over the operator basis, leading to a
 complete scrambling of the information in the quantum system.