We clarify the lore that anomaly-free symmetries are either on-site or can be transformed into on-site symmetries. We prove that any finite, internal, anomaly-free symmetry in a $1+1\text{D}$ lattice Hamiltonian system can be disentangled into an on-site symmetry by introducing ancillae and applying conjugation via a finite-depth quantum circuit. We provide an explicit construction of the disentangling circuit using Gauss’s law operators and emphasize the necessity of adding ancillae. Our result establishes the converse to a generalized Lieb-Schultz-Mattis theorem by demonstrating that any anomaly-free symmetry admits a trivially gapped Hamiltonian.

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