Phases of matrix-product states with symmetries and measurements: Finite nilpotent groups

Abstract

We classify phases of one-dimensional matrix-product states (MPS) under symmetric circuits augmented with symmetric measurements and feedforward. Building on the framework introduced in Gunn et al., Phys. Rev. B 111, 115110 (2025), we extend the analysis from abelian and class-2 nilpotent groups to all finite nilpotent groups. For any such symmetry group $G$, we construct explicit protocols composed of $G$-symmetric circuits and measurements with feedforward that transform symmetry-protected topological (SPT) states into the trivial phase and vice versa using a finite number of measurement rounds determined by the nilpotency class of $G$. Although these transformations are approximate, we prove that their success probability converges to unity in the thermodynamic limit, establishing asymptotically deterministic equivalence. Consequently, all SPT phases protected by finite nilpotent groups collapse to a single phase once symmetric measurements and feedforward are allowed. We further show that the same holds for non-normal MPS with long-range correlations, including GHZ-type states. The central technical ingredient is a hierarchical structure of irreducible representations of nilpotent groups, which enables a recursive reduction of non-abelian components to abelian ones. Our results demonstrate that symmetric measurements lead to a complete collapse of both symmetry-protected and non-normal MPS phases for all finite nilpotent symmetry groups.