Dispersion Bound on Krylov Complexity

Starting from the generalized Schrödinger uncertainty principle for superoperators in Krylov space, we formulate a universal bound on the rate of complexity growth in isolated systems [1]. Remarkably, saturation occurs when the observables follow the trajectory of the generalized coherent states of the complexity algebra. We identify the three possible scenarios of maximal complexity growth and discuss how operator growth deviates from the speed limit in generic systems. Finally, we show that chaotic behavior is not necessary for the saturation of the dispersion bound. Conversely, in systems governed by random Hamiltonians, a paradigmatic model of quantum chaos, Krylov complexity does not evolve at the maximal speed limit.