Generating Einsteinx2013Podolskyx2013Rosen correlations for teleporting collective spin states in a two dimensional trapped ion crystal

We propose the use of phononx2013mediated interactions as an entanglement resource to engineer Einsteinx2013Podolskyx2013Rosen (EPR) correlations and to perform teleportation of collective spin states in twox2013dimensional 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.