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We study the scrambling of local quantum information in chaotic many-bodysystems in the presence of a locally conserved quantity like charge or energythat moves diffusively. The interplay between conservation laws and scramblingsheds light on the mechanism by which unitary quantum dynamics, which isreversible, gives rise to diffusive hydrodynamics, which is a dissipativeprocess. We obtain our results in a random quantum circuit model that isconstrained to have a conservation law. We find that a generic spreadingoperator consists of two parts: (i) a conserved part which comprises the weightof the spreading operator on the local conserved densities, whose dynamics isdescribed by diffusive charge spreading. This conserved part also acts as asource that steadily emits a flux of (ii) non-conserved operators. Thisemission leads to dissipation in the operator hydrodynamics, with thedissipative process being the conversion of operator weight from localconserved operators to nonconserved, at a rate set by the local diffusioncurrent. The emitted nonconserved parts then spread ballistically at abutterfly speed, thus becoming highly nonlocal and hence essentiallynon-observable, thereby acting as the "reservoir" that facilitates thedissipation. In addition, we find that the nonconserved component develops apower law tail behind its leading ballistic front due to the slow dynamics ofthe conserved components. This implies that the out-of-time-order commutator(OTOC) between two initially separated operators grows sharply upon the arrivalof the ballistic front but, in contrast to systems with no conservation laws,it develops a diffusive tail and approaches its asymptotic late-time value onlyas a power of time instead of exponentially. We also derive these resultswithin an effective hydrodynamic description which contains multiple coupleddiffusion equations.

Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws