Perfect State Transfer from a Localised Two-Excitation State to a Dicke State via Static Spin-Network Hamiltonians

This paper addresses the problem of engineering a time-independent, excitation-preserving Hamiltonian that can perform perfect state transfer (PST) between two specific quantum states: a localized two-excitation state (where excitations are on two distinct sites) and the symmetric two-excitation Dicke state. While PST between arbitrary real states is theoretically possible with unconstrained symmetric matrices, the author seeks a solution that adheres to the physical constraints of a spin-network Hamiltonian.

The author exploits the $S_{N-2}$ permutation symmetry of the initially unoccupied sites to reduce the dynamics of the two-excitation sector to a four-dimensional invariant subspace. By requiring a specific linear combination of the initial and target states to be a zero eigenvector of the Hamiltonian, the author determines the on-site energies in closed form. The remaining inverse spectral problem is solved by reducing it to a reciprocal sextic polynomial, which is further simplified to a cubic. The existence of a real solution is proven symbolically using the intermediate value theorem, without relying on numerical optimization.

The study proves that for every system size $N \ge 4$, there exists a real, time-independent Hamiltonian that realizes perfect state transfer from the localized state to the Dicke state at a finite time. This construction is entirely symbolic and avoids the pitfalls of numerical optimization, which often fails to find exact solutions in the constrained parameter space of spin networks. The author provides a worked example for $N=6$ and validates the construction numerically for system sizes up to $N=102$, showing that the residuals for state transfer are at machine precision.

This result provides a deterministic, static-Hamiltonian method for preparing Dicke states, which are essential resources in quantum information processing, particularly for quantum sensing and entanglement distribution. Unlike circuit-based or time-dependent control schemes, this approach relies on a fixed, engineered interaction, offering a distinct pathway for state preparation in quantum spin networks.