Intrinsic symmetry-protected topological mixed state from modulated symmetries and hierarchical structure of boundary anomaly

We introduce a class of intrinsic symmetry-protected topological mixed state (mSPT) in open quantum systems that feature modulated symmetries, such as dipole and subsystem symmetries. Intriguingly, these mSPT phases cannot be realized as the ground states of a gapped Hamiltonian under thermal equilibrium. The microscopic form of the density matrix characterizing these intrinsic mixed-state SPT ensembles is constructed using solvable coupled-wire models that incorporate quenched disorder or quantum channels. A detailed comparison of the hierarchical structure of boundary anomalies in both pure and mixed states is presented, utilizing flux insertion and Laughlin's charge pumping arguments. Finally, we explore the salient features of boundary anomalies in the mixed-state ensemble, which can be detected through the R\'enyi-$N$ correlation function of charged observables.