Asymptotic behavior near projective no-click postselection

Determine the asymptotic behavior of the characteristic length and the associated weak-symmetry spontaneous symmetry-breaking correlations in the no-click postselected one-dimensional cluster-state ASPT as the measurement strength approaches the projective limit q_x=1.

Background

The paper also examines symmetry-preserving no-click postselection generated by weak X measurements on odd sites, with measurement strength q_x. This operation competes with the conserved stabilizers responsible for the ASPT Renyi-2 string order and enhances purity-weighted ZZ correlations associated with weak-symmetry spontaneous symmetry breaking.

The numerical data show a rapidly increasing characteristic length as q_x approaches the projective limit q_x=1, but do not provide evidence for a finite-q_x critical point. Because the available system sizes cannot resolve the limiting scaling, the asymptotic behavior of the correlation length and correlations remains unresolved.

References

Rather, it is consistent with a rapidly increasing characteristic length as the projective limit $q_x\to 1$ is approached, although the available system sizes do not allow us to determine its asymptotic behavior.

— Fate of average symmetry-protected topological states under symmetry-preserving quantum operations  (2609.27530 - Kuno et al., 23 Sep 2026) in Section VI, subsection “No-click postselection”