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Symmetry protected topological phases exhibit nontrivial short-rangedentanglement protected by symmetry and cannot be adiabatically connected totrivial product states while preserving the symmetry. In contrast, intrinsictopological phases do not need ordinary symmetry to stabilize them and theirground states exhibit long-range entanglement. It is known that for a givensymmetry group $G$, the 2D SPT phase protected by $G$ is dual to the 2Dtopological phase exemplified by the twisted quantum double model$D^{\omega}(G)$ via gauging the global symmetry $G$. Recently it was realizedthat such a general gauging map can be implemented by some local unitaries andlocal measurements when $G$ is a finite, solvable group. Here, we review thegeneral approach to gauging a $G$-SPT starting from a fixed-point ground-statewave function and applying a $N$-step gauging procedure. We provide an in-depthanalysis of the intermediate states emerging during the N-step gauging andprovide tools to measure and identify the emerging symmetry-enrichedtopological order of these states. We construct the generic lattice parentHamiltonians for these intermediate states, and show that they form anentangled superposition of a twisted quantum double with an SPT ordered state.Notably, we show that they can be connected to the TQD through a finite-depth,local quantum circuit which does not respect the global symmetry of the SETorder. We introduce the so-called symmetry branch line operators and show thatthey can be used to extract the symmetry fractionalization classes and symmetrydefectification classes of the SET phases with the input data $G$ and$[\omega]\in H^3(G,U(1))$ of the pre-gauged SPT ordered state. We illustratethe procedure of preparing and characterizing the emerging SET ordered statesfor some Abelian and non-Abelian examples such as dihedral groups $D_n$ and thequaternion group $Q_8$.

Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements