Dynamics of R\'enyi entanglement entropy in diffusive qudit systems

My previous work [arXiv:1902.00977] studied the dynamics of R'enyi entanglement entropy $R_{\alpha }$ in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that $R_{\alpha }$ with R'enyi index $\alpha >1$ grows no faster than “diffusively” (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-$1\mathrm{/}2$ systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension $d\ge 2$.