Operational criterion for controlled dense coding with non-trivial tripartite entangled states

In this paper, we provide an operational criterion for controlled dense coding (CDC) with a general class of three-qubit partially entangled states. A general three-qubit pure entangled state can be classified into two inequivalent classes according to their genuine tripartite entanglement. We claim that if a three-qubit state shows entanglement characteristic similar to Greenberger–Horne–Zeilinger (GHZ)-class, then such non-trivial tripartite states are useful in CDC whereas states belonging to the W-class are not useful for that. We start with a particular class of non-trivial partially entangled states belonging to the GHZ-class and show that they are effective in CDC. Then we cite several other examples of different types of tripartite entangled states to support our conjecture.