Anomaly-Enforced Long-Range Correlations in Measured Symmetry Protected Topological States

Topological cluster states—canonical resource states for measurement-based quantum computation—motivate us to ask what local symmetry-charge measurements reveal in generic bosonic G×H Symmetry Protected Topological (SPT) States with abelian G and H. We show that, upon measuring local G charges, post-measurement trajectories with uniform outcomes admit a space–time rotation that maps their correlators to those of an SPT boundary. The boundary’s ’t Hooft anomaly then enforces either long-range order or critical power-law decay. Beyond post-selection, we analyze trajectory-averaged higher moments via a replica trick that couples copies of the SPT state along a symmetric defect; the same space-time rotation map and anomaly argument implies long-range correlations in the conditioned ensemble. We substantiate these claims in a concrete example of 2D Z₂×Z₂×Z₂ (type-III) SPT under the measurement of one Z₂ charge. Finally, we outline a measurement-and-feedback protocol that enables extraction of critical correlations from observables linear in the density matrix, offering a practical route to experimental diagnostics.