Optimal dense coding using a partially-entangled pure state of Schmidt rank
DΛ and a noiseless quantum channel of dimension D is studied both in
the deterministic case where at most Ldβ messages can be transmitted with
perfect fidelity, and in the unambiguous case where when the protocol succeeds
(probability Οxβ) Bob knows for sure that Alice sent message x, and when
it fails (probability 1βΟxβ) he knows it has failed. Alice is allowed any
single-shot (one use) encoding procedure, and Bob any single-shot measurement.
For DΛβ€D a bound is obtained for Ldβ in terms of the largest
Schmidt coefficient of the entangled state, and is compared with published
results by Mozes et al. For DΛ>D it is shown that Ldβ is strictly
less than D2 unless DΛ 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 DΛβ€D, assuming Οxβ>0 for a
set of DΛD 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 DΛ=D) that are
orthogonal for different messages. Both bounds are saturated when Οxβ is a
constant independent of x, by a protocol based on one-shot entanglement
concentration. For DΛ>D it is shown that (at least) D2 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.Comment: Short new section VII added. Latex 23 pages, 1 PSTricks figure in
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