  We consider the problem of communicating quantum states by simultaneously
making use of a noiseless classical channel, a noiseless quantum channel and
shared entanglement. We specifically study the version of the problem in which
the sender is given knowledge of the state to be communicated. In this setting,
a trade-off arises between the three resources, some portions of which have
been investigated previously in the contexts of the quantum-classical trade-off
in data compression, remote state preparation and superdense coding of quantum
states, each of which amounts to allowing just two out of these three
resources. We present a formula for the triple resource trade-off that reduces
its calculation to evaluating the data compression trade-off formula. In the
process, we also construct protocols achieving all the optimal points. These
turn out to be achievable by trade-off coding and suitable time-sharing between
optimal protocols for cases involving two resources out of the three mentioned
above.

Quantum Physics