Family-Vicsek (FV) scaling provides an understanding for the growth and finite-size saturation of fluctuations in classical systems. Here, we extend the FV roughness to transferred segment magnetization after quantum quenches in a dissipative XXZ spin chain with homogeneous gain and loss, starting from a nonequilibrium steady state with finite magnetization. In the noninteracting limit we derive a closed-form expression for the roughness in the presence of dissipation. It displays two-parameter FV scaling and smoothly interpolates between the clean ballistic behavior and the dissipation-dominated scalings. For interacting chains, tensor-network simulations show that the nondissipative ballistic growth at finite magnetization is robust, whereas the full Lindblad evolution is generically controlled by the dissipative relaxation time and exhibits a dissipation-dominated collapse.

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• Dissipative dynamics
• Scaling methods
• XYZ model