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Optimal dense coding using a partially-entangled pure state of Schmidt rank 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 τ x ) Bob knows for sure that Alice sent message x , and when it fails (probability 1 − τ x ) he knows it has failed. Alice is allowed any single-shot (one use) encoding procedure, and Bob any single-shot measurement. For D ¯ ⩽ 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 Phys. Rev. A 71 , 012311 ( 2005 ) . For D ¯ > D it is shown that L d is strictly less than D 2 unless 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 D ¯ ⩽ D , assuming τ x > 0 for a set of D ¯ D messages, and a bound is obtained for the average ⟨ 1 ∕ τ ⟩ . A bound on the average ⟨ τ ⟩ requires an additional assumption of encoding by isometries (unitaries when D ¯ = D ) that are orthogonal for different messages. Both bounds are saturated when τ x is a constant independent of x , by a protocol based on one-shot entanglement concentration. For 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.
Physical Review A – American Physical Society (APS)
Published: Apr 1, 2006
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