Scalable High-Dimensional Multipartite Entanglement with Trapped Ions

We propose a protocol for the preparation of generalized Greenberger-Horne-Zeilinger (GHZ) states of N atoms each with d=3 or 4 internal levels. We generalize the celebrated one-axis twisting (OAT) Hamiltonian for N qubits to qudits by including OAT interactions of equal strengths between every pair of qudit levels, a protocol we call as balanced OAT (BOAT). Analogous to OAT for qubits, we find that starting from a product state of an arbitrary number of atoms N, dynamics under BOAT leads to the formation of GHZ states for qutrits $d=3$ and ququarts $d=4$. While BOAT could potentially be realized on several platforms where all-to-all coupling is possible, here we propose specific implementations using trapped ion systems. We show that preparing these states with a fidelity above a threshold value rules out lower dimensional entanglement than that of the generalized GHZ states. For qutrits, we also propose a protocol to bound the fidelity that requires only global addressing of the ion crystal and single-shot readout of one of the levels. Our results open a path for the scalable generation and certification of high-dimensional multipartite entanglement on current atom-based quantum hardware.