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

TL;DR

This paper classifies phases of one-dimensional matrix-product states with finite nilpotent symmetries under symmetric local circuits, showing all such phases collapse into a single phase via symmetric measurements and feedforward.

Contribution

It extends the classification of G-CMF phases to all finite nilpotent groups and constructs explicit protocols to trivialize complex phases.

Findings

Complete classification of G-CMF phases for all finite nilpotent groups.

Explicit symmetry-respecting protocols for phase trivialization.

All SPT and non-normal phases collapse into a single phase under G-CMF transformations.

Abstract

We study phases of one-dimensional matrix-product states (MPS) when transformations are restricted to symmetric local circuits supplemented with symmetric measurements and feedforward (G-CMF). Building on the framework introduced in Gunn et al., Phys. Rev. B 111, 115110 (2025), we extend the analysis to all finite nilpotent groups for which we obtain a complete classification of G-CMF phases. We construct explicit symmetry-respecting protocols that map any symmetry-protected topological (SPT) or non-normal (GHZ-type) MPS to the trivial phase-and vice versa-with success probability approaching one in the thermodynamic limit. The key technical ingredient is a finite hierarchical structure of irreducible representations of nilpotent groups, which enables successive rounds of symmetric measurements to systematically reduce non-abelian components to abelian ones. Our results demonstrate that allowing symmetric measurements and feedforward fundamentally simplifies the phase structure of 1D systems with nilpotent symmetries: all SPT and non-normal MPS phases collapse into a single asymptotically equivalent phase under G-CMF transformations.