Fate of average symmetry-protected topological states under symmetry-preserving quantum operations

Abstract

The fate of average symmetry-protected topological (ASPT) states based on the one-dimensional cluster state is investigated by using the Choi mapping, namely, the doubled Hilbert-space formalism. By introducing two effective spins, $τ$ and $η$, on each doubled rung, we obtain a transparent representation of the ASPT parent Hamiltonian and systematically identify the conserved stabilizers and the corresponding orders. We further study symmetry-preserving operations that deform the ASPT by combining qualitative effective-Hamiltonian and systematic parent-Hamiltonian analyses with numerical matrix-product-state filtering calculations. We find that although the ASPT remains robust over a broad range of decoherence strengths, strong-to-weak spontaneous symmetry breaking (SWSSB) becomes exact in the maximal-decoherence limit. In addition, we consider symmetry preserving no-click postselection, which removes the exact conservation law supporting the Rényi-2 string order of the ASPT and drives the system toward weak-symmetry spontaneous symmetry breaking (WSSB). Again, the ASPT remains robust over a broad range of postselection strengths, with WSSB becoming pronounced only near the projective limit and exact at the limit itself.

The paper asks how the average symmetry-protected topological (ASPT) state obtained by even-site $Z$ dephasing of a one-dimensional cluster state responds to further operations preserving its remaining strong and weak symmetries. It studies even-site $X$ dephasing, with probability $0\le p_x\le 1/2$, and postselection of positive outcomes from odd-site weak $X$ measurements, with strength $0\le q_x\le1$. Using Choi vectorization, a local mapping to two effective spins per doubled rung identifies exactly conserved cluster stabilizers and relates them to Rényi-2 and purity-weighted string correlators. Filtering transformations construct exact parent Hamiltonians. Simplified Hamiltonians suggest Ising-like competition between ASPT order and symmetry breaking, but the exact constructions contain additional correlated interactions; for postselection, these already appear at the same leading order as the simplified perturbation. Projector constraints establish exact strong-to-weak spontaneous symmetry-breaking correlations at maximal dephasing and exact weak-symmetry-breaking correlations at projective postselection. DMRG preparation and matrix-product-state filtering, evaluated at initial dephasing $p_z=0.2$, show string-order suppression and enhanced symmetry-breaking correlations concentrated near those endpoints. Finite-size changes shift toward stronger operations as system size increases. Fitted characteristic lengths reach approximately $10^3$ at $p_x=0.442$ and $2.8\times10^3$ at $q_x=0.940$. The accessible sizes provide no clear evidence for finite-strength critical points and do not determine the asymptotic length divergence. The analysis is restricted to the one-dimensional cluster construction; extension to higher dimensions remains open.