Generating Einstein$\text{unicode}x2013$Podolsky$\text{unicode}x2013$Rosen correlations for teleporting collective spin states in a two dimensional trapped ion crystal

We propose the use of phonon$\text{unicode}x2013$mediated interactions as an entanglement resource to engineer Einstein$\text{unicode}x2013$Podolsky$\text{unicode}x2013$Rosen (EPR) correlations and to perform teleportation of collective spin states in two$\text{unicode}x2013$dimensional ion crystals. We emulate continuous variable quantum teleportation protocols between subsystems corresponding to different nuclear spin degrees of freedom. In each of them, a quantum state is encoded in an electronic spin degree of freedom that couples to the vibrational modes of the crystal. We show that high fidelity teleportation of spin-coherent states and their phase-displaced variant, entangled spin-squeezed states, and Dicke states, is possible for realistic experimental conditions in arrays from a few tens to a few hundred ions.