  We use non-equilibrium steady states to study the effect of
dissipation-assisted operator evolution (DAOE) on the scaling behavior of
transport in one-dimensional spin chains. We consider three models in the XXZ
family: the XXZ model with staggered anisotropy, which is chaotic; XXZ model
with no external field and tunable interaction, which is Bethe ansatz
integrable and (in the zero interaction limit) free fermion integrable; and the
disordered XY model, which is free-fermion integrable and Anderson localized.
We find evidence that DAOE's effect on transport is controlled by its effect on
the system's conserved quantities. To the extent that DAOE preserves those
symmetries, it preserves the scaling of the system's transport properties; to
the extent it breaks those conserved quantities, it pushes the system towards
diffusive scaling of transport.
