A note on invariants of mixed-state topological order in 2D

The classification of mixed-state topological order requires indices that behave monotonically under finite-depth quantum channels. In two dimensions, a braided $C^*$-tensor category, which corresponds to strong symmetry, arises from a state satisfying approximate Haag duality. In this note, we show that the $S$-matrix and topological twists of the braided $C^*$-tensor category are quantities that are monotone under finite-depth quantum channels.