Abstract
Optimal dense coding using a partially-entangled pure state of Schmidt rank and a noiseless quantum channel of dimension is studied both in the deterministic case where at most messages can be transmitted with perfect fidelity, and in the unambiguous case where when the protocol succeeds (probability ) Bob knows for sure that Alice sent message , and when it fails (probability ) he knows it has failed. Alice is allowed any single-shot (one use) encoding procedure, and Bob any single-shot measurement. For a bound is obtained for in terms of the largest Schmidt coefficient of the entangled state, and is compared with published results by Mozes et al. [Phys. Rev. A71, 012311 (2005)]. For it is shown that is strictly less than unless is an integer multiple of , in which case uniform (maximal) entanglement is not needed to achieve the optimal protocol. The unambiguous case is studied for , assuming for a set of messages, and a bound is obtained for the average . A bound on the average requires an additional assumption of encoding by isometries (unitaries when ) that are orthogonal for different messages. Both bounds are saturated when is a constant independent of , by a protocol based on one-shot entanglement concentration. For it is shown that (at least) 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.
- Received 29 December 2005
DOI:https://doi.org/10.1103/PhysRevA.73.042311
©2006 American Physical Society