  The quantum advantage of dense coding is studied, considering general
encoding quantum operations. Particular attention is devoted to the case of
many senders, and it is shown that restrictions on the possible operations on
the senders' side may make some quantum state useless for dense-coding. It is
shown, e.g., that some states are useful for dense coding if the senders can
communicate classically (but not quantumly), yet they cannot be used for dense
coding, if classical communication is not allowed. These no-go results are
actually independent of the particular quantification of the quantum advantage,
being valid for any reasonable choice. It is further shown that the quantum
advantage of dense coding satisfies a monogamy relation with the so-called
entanglement of purification.
