Deterministic and unambiguous dense coding

Abstract

Optimal dense coding using a partially-entangled pure state of Schmidt rank $\overline{D}$ 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 ${\ensuremath{\tau}}_{x}$) Bob knows for sure that Alice sent message $x$, and when it fails (probability $1\ensuremath{-}{\ensuremath{\tau}}_{x}$) he knows it has failed. Alice is allowed any single-shot (one use) encoding procedure, and Bob any single-shot measurement. For $\overline{D}\ensuremath{\leqslant}D$ 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. A71, 012311 (2005)]. For $\overline{D}>D$ it is shown that ${L}_{d}$ is strictly less than ${D}^{2}$ unless $\overline{D}$ 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 $\overline{D}\ensuremath{\leqslant}D$, assuming ${\ensuremath{\tau}}_{x}>0$ for a set of $\overline{D}D$ messages, and a bound is obtained for the average $⟨1∕\ensuremath{\tau}⟩$. A bound on the average $⟨\ensuremath{\tau}⟩$ requires an additional assumption of encoding by isometries (unitaries when $\overline{D}=D$) that are orthogonal for different messages. Both bounds are saturated when ${\ensuremath{\tau}}_{x}$ is a constant independent of $x$, by a protocol based on one-shot entanglement concentration. For $\overline{D}>D$ 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.