  We study a continuous variable (CV) dense-coding protocol, originally
proposed to employ a two-mode squeezed state, using a general two-mode Gaussian
state as a quantum channel. We particularly obtain conditions to manifest
quantum advantage by beating two well-known single-mode schemes, namely, the
squeezed-state scheme (best Gaussian scheme) and the number-state scheme
(optimal scheme achieving the Holevo bound). We then extend our study to a
multipartite Gaussian state and investigate the monogamy of operational
entanglement measured by the communication capacity under the dense-coding
protocol. We show that this operational entanglement represents a strict
monogamy relation, by means of Heisenberg's uncertainty principle among
different parties, i.e., the quantum advantage for communication can be
possible for only one pair of two-mode systems among many parties.
