Deterministic and unambiguous dense coding

Optimal dense coding using a partially-entangled pure state of Schmidt rank D̅ 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 D̅ ⩽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. Phys. Rev. A 71 012311 (2005)]. For D̅ >D it is shown that L_d is strictly less than D² unless D̅ 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 D̅ ⩽D, assuming τ_x>0 for a set of D̅ 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 D̅ =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 D̅ >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.