Generation of hyperentangled states between remote noninteracting atomic ions

We propose a scheme of generating four-qubit hyperentangled states between a pair of remote noninteracting atomic ions with a Λ configuration that are confined in Paul traps. These hyperentangled states, different from the normal entangled states that are entangled in a single degree of freedom, are entangled in both spin and motion degrees of freedom. In our proposal, the entanglement is first generated in spin degrees of freedom using linear optics and then transferred to the motion degree of freedom using a sequence of laser pluses, including the stimulated Raman carrier transitions and sideband transitions. The proposal is completed with regenerating entanglement in spin degrees of freedom using linear optics.