Generation and classification of robust remote symmetric Dicke states

In this paper, we present an approach to generating arbitrary symmetric Dicke states with distant trapped ions and linear optics. Distant trapped ions can be prepared in the symmetric Dicke states by using two photon-number-resolving detectors and a polarization beam splitter. The atomic symmetric Dicke states are robust against decoherence, for atoms are in a metastable level. We discuss the experimental feasibility of our scheme with current technology. Finally, we discuss the classification of arbitrary $n$-qubit symmetric Dicke states under statistical local operation and classical communication and prove the existence of $n/2]$ inequivalent classes of genuine entanglement of $n$-qubit symmetric Dicke states.