Generating Einstein--Podolsky--Rosen correlations for teleporting collective spin states in a two dimensional trapped ion crystal

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