  Optimal dense coding using a partially-entangled pure state of Schmidt rank
$\bar D$ and a noiseless quantum channel of dimension $D$ is studied both in
the deterministic case where at most $L_d$ messages can be transmitted with
perfect fidelity, and in the unambiguous case where when the protocol succeeds
(probability $\tau_x$) Bob knows for sure that Alice sent message $x$, and when
it fails (probability $1-\tau_x$) he knows it has failed. Alice is allowed any
single-shot (one use) encoding procedure, and Bob any single-shot measurement.
For $\bar D\leq D$ a bound is obtained for $L_d$ in terms of the largest
Schmidt coefficient of the entangled state, and is compared with published
results by Mozes et al. For $\bar D > D$ it is shown that $L_d$ is strictly
less than $D^2$ unless $\bar D$ is an integer multiple of $D$, in which case
uniform (maximal) entanglement is not needed to achieve the optimal protocol.
The unambiguous case is studied for $\bar D \leq D$, assuming $\tau_x>0$ for a
set of $\bar D D$ messages, and a bound is obtained for the average
$\lgl1/\tau\rgl$. A bound on the average $\lgl\tau\rgl$ requires an additional
assumption of encoding by isometries (unitaries when $\bar D=D$) that are
orthogonal for different messages. Both bounds are saturated when $\tau_x$ is a
constant independent of $x$, by a protocol based on one-shot entanglement
concentration. For $\bar D > D$ it is shown that (at least) $D^2$ messages can
be sent unambiguously. Whether unitary (isometric) encoding suffices for
optimal protocols remains a major unanswered question, both for our work and
for previous studies of dense coding using partially-entangled states,
including noisy (mixed) states.
