Quantum teleportation of many-body spin states in a two-dimensional trapped ions crystal

We propose to use phonon-mediated interactions as an entanglement resource to perform teleportation of many-body spin states. The spin ensembles correspond to different nuclear spin degrees of freedom of a two-dimensional ion crystal created in a Penning trap. In each ensemble, an electronic spin degree of freedom is used to encode a qubit that couples to the vibrational mode of the crystal. The strong magnetic field native to the Penning trap generates large enough energy splittings for the preparation and read out of each of the different spin ensembles without affecting the others. Emulating continuous variable quantum teleportation protocols that rely on two-mode-squeezing and beam-splitting operations, we theoretically discuss how to engineer suitable effective spin-spin interactions between three spin ensembles to implement a teleportation circuit. Exact numerical simulations of the circuit confirm the protocol performs teleportation of the many-body spin state with good fidelity for realistic experimental conditions in arrays from a few tens to a few hundred ions. Our analysis demonstrates that teleportation of spin-coherent and entangled spin-squeezed states as well as their phase-displaced versions is possible in this setting and opens new opportunities for quantum information processing in a Penning trap where single ion resolution is not trivial.