  Entanglement-assisted classical communication and transmission of a quantum
system are the two quantum resources for information processing. Many
information tasks can be performed using either quantum resource. However, this
equivalence is not always present since entanglement assisted classical
communication is known to sometimes be the better performing resource. Here, we
show not only the opposite phenomenon; that there exists tasks for which
transmission of a quantum system is a more powerful resource than entanglement
assisted classical communication, but also that such phenomena can have a
surprisingly strong dependence on the dimension of Hilbert space. We introduce
a family of communication complexity problems parametrized by dimension of
Hilbert space and study the performance of each quantum resource. We find that
for low dimensions, the two resources perform equally well, whereas for
dimension seven and above, the equivalence is suddenly broken and transmission
of a quantum system becomes more powerful than entanglement assisted classical
communication. Moreover, we find that transmission of a quantum system may even
outperform classical communication assisted by the stronger-than-quantum
correlations obtained from the principle of Macroscopic Locality.
