Extension to higher-excitation Dicke states

Determine whether a different static-Hamiltonian architecture can extend perfect transfer from a localised higher-excitation state to the corresponding Dicke state for excitation numbers k≥3.

Background

The paper constructs a real, time-independent, excitation-preserving spin-network Hamiltonian that transfers the localised two-excitation state |110⋯0⟩ to the two-excitation Dicke state for every system size N≥4. The construction relies on an S_{N−2} symmetry reduction to a four-dimensional invariant subspace and on a zero-mode condition that leaves sufficient spectral freedom.

For excitation numbers k≥3, the paper explains that the analogous S_{N−k} reduction and x₊ zero-mode architecture encounter parameter-count and diagonal-surjectivity limitations. However, these observations are explicitly described as obstructions to that particular architecture rather than as a no-go theorem for all static-Hamiltonian constructions. The unresolved question is therefore whether another architecture can achieve the corresponding higher-excitation transfer.