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The classification of topological phases of matter is fundamental tounderstand and characterize the properties of quantum materials. In this paperwe study phases of matter in one-dimensional open quantum systems. We definetwo mixed states to be in the same phase if both states can be transformed intothe other by a shallow circuit of local quantum channels. We aim to understandthe phase diagram of matrix product density operators that are renormalizationfixed points. These states arise, for example, as boundaries of two-dimensionaltopologically ordered states. We first construct families of such states basedon C*-weak Hopf algebras, the algebras whose representations form a fusioncategory. More concretely, we provide explicit local fine-graining and localcoarse-graining quantum channels for the renormalization procedure of thesestates. Finally, we prove that those arising from C*-Hopf algebras are in thetrivial phase.

Matrix Product Operator Algebras II: Phases of Matter for 1D Mixed States