Hierarchy of emergent cluster states by measurement from symmetry-protected-topological states with large symmetry to a subsystem cat state

We propose a measurement-producing hierarchy emerging among correlated states by sequential subsystem projective measurements. We start from symmetry-protected-topological (SPT) cluster states with a large symmetry and apply sequential subsystem projective measurements to them and find that generalized cluster SPT states with a reduced symmetry appear in the subsystem of the unmeasured sites. That prescription finally produces Greenberger-Horne-Zeilinger states with long-range order in the subsystem composed of periodic unmeasured sites of the original lattice. The symmetry-reduction hierarchical structure from a general large-symmetric SPT cluster state is clearly captured by the measurement update flow in the efficient algorithm of stabilizer formalism. This approach is useful not only for the analytical search for the measured state, but also for numerical simulation with a large system size. We also numerically verify the symmetry-reduction hierarchy by sequential subsystem projective measurements applied to large systems and large-symmetric cluster SPT states.