  In order to understand the resourcefulness of a natural quantum system in
quantum communication tasks, we study the dense coding capacity (DCC) and
teleportation fidelity (TF) of Haar uniformly generated random multipartite
states of various ranks. We prove that when a rank-2 two-qubit state, a Werner
state, and a pure state possess the same amount of entanglement, the DCC of a
rank-2 state belongs to the envelope made by pure and Werner states. In a
similar way, we obtain an upper bound via the generalized
Greenberger-Horne-Zeilinger state for rank-2 three-qubit states when the dense
coding with two senders and a single receiver is performed and entanglement is
measured in the senders:receiver bipartition. The normalized frequency
distribution of DCC for randomly generated two-, three- and four-qubit density
matrices with global as well as local decodings at the receiver's end are
reported. The estimation of mean DCC for two-qubit states is found to be in
good agreement with the numerical simulations. Universally, we observe that the
performance of random states for dense coding as well as teleportation
decreases with the increase of the rank of states which we have shown to be
surmounted by the local pre-processing operations performed on the shared
states before starting the protocols, irrespective of the rank of the states.
The local pre-processing employed here is based on positive operator valued
measurements along with classical communication and we show that unlike dense
coding with two-qubit random states, senders' operations are always helpful to
probabilistically enhance the capabilities of implementing dense coding as well
as teleportation.
