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

We propose the use of phonon$\unicode{x2013}$mediated interactions as an entanglement resource to engineer Einstein$\unicode{x2013}$Podolsky$\unicode{x2013}$Rosen (EPR) correlations and to perform teleportation of collective spin states in two$\unicode{x2013}$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.