We present a rigorous proof of an interesting boundary effect of deterministic dense coding first observed by S. Mozes, J. Oppenheim, and B. Reznik [Phys. Rev. A 71, 012311 (2005)]. Namely, it is shown that ${d}^{2}\ensuremath{-}1$ cannot be the maximal alphabet size of any isometric deterministic dense coding schemes utilizing $d$-level partial entanglement.
We present a rigorous proof of an interesting boundary effect of deterministic dense coding first observed by S. Mozes, J. Oppenheim, and B. Reznik [Phys. Rev. A 71, 012311 (2005)]. Namely, it is shown that d2\ensuremath−1 cannot be the maximal alphabet size of any isometric deterministic dense coding schemes utilizing d-level partial entanglement.
The utilization of a d-level partially entangled state, shared by two parties wishing to communicate classical information without errors over a noiseless quantum channel, is discussed. We analytically construct deterministic dense coding schemes for certain classes of nonmaximally entangled states, and numerically obtain schemes in the general case. We study the dependency of the maximal alphabet size of such schemes on the partially entangled state shared by the two parties. Surprisingly, for d>2 it is possible to have deterministic dense coding with less than one ebit. In this case the number of alphabet letters that can be communicated by a single particle is between d and 2d. In general, we numerically find that the maximal alphabet size is any integer in the range [d,d2] with the possible exception of ${d}^{2}\ensuremath{-}1$. We also find that states with less entanglement can have a greater deterministic communication capacity than other more entangled states.
We present an analytical study of the standard two-party deterministic dense-coding protocol, under which communication of perfectly distinguishable messages takes place via a qudit from a pair of nonmaximally entangled qudits in a pure state $|\ensuremath{\psi}⟩$. Our results include the following: (i) We prove that it is possible for a state $|\ensuremath{\psi}⟩$ with lower entanglement entropy to support the sending of a greater number of perfectly distinguishable messages than one with higher entanglement entropy, confirming a result suggested via numerical analysis in Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. (ii) By explicit construction of families of local unitary operators, we verify, for dimensions d=3 and d=4, a conjecture of Mozes et al. about the minimum entanglement entropy that supports the sending of d+j messages, $2\ensuremath{\le}j\ensuremath{\le}d\ensuremath{-}1$; moreover, we show that the j=2 and $j=d\ensuremath{-}1$ cases of the conjecture are valid in all dimensions. (iii) Given that $|\ensuremath{\psi}⟩$ allows the sending of K messages and has $\sqrt{{\ensuremath{\lambda}}_{0}}$ as its largest Schmidt coefficient, we show that the inequality ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$, established by Wu et al. [Phys. Rev. A 73, 042311 (2006)], must actually take the form ${\ensuremath{\lambda}}_{0}<d/K$ if K=d+1, while our constructions of local unitaries show that equality can be realized if K=d+2 or $K=2d\ensuremath{-}1$.
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 Ld 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 Ld 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 D>D it is shown that Ld is strictly less than D2 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 $\overline{D}\ensuremath{\leqslant}D$, assuming ${\ensuremath{\tau}}_{x}>0$ for a set of DD 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 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 D>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.
Under two-party deterministic dense coding, Alice communicates (perfectly distinguishable) messages to Bob via a qudit from a pair of entangled qudits in pure state $|\ensuremath{\Psi}\ensuremath{\rangle}$. If $|\ensuremath{\Psi}\ensuremath{\rangle}$ represents a maximally entangled state (i.e., each of its Schmidt coefficients is 1/d), then Alice can convey to Bob one of d2 distinct messages. If $|\ensuremath{\Psi}\ensuremath{\rangle}$ is not maximally entangled, then Ji et al. [Phys. Rev. A 73, 034307 (2006)] have shown that under the original deterministic dense-coding protocol, in which messages are encoded by unitary operations performed on Alice's qudit, it is impossible to encode ${d}^{2}\ensuremath{-}1$ messages. Encoding ${d}^{2}\ensuremath{-}2$ messages is possible; see, for example, the numerical studies by Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. Answering a question raised by Wu et al. [Phys. Rev. A 73, 042311 (2006)], we show that when $|\ensuremath{\Psi}\ensuremath{\rangle}$ is not maximally entangled, the communications limit of ${d}^{2}\ensuremath{-}2$ messages persists even when the requirement that Alice encode by unitary operations on her qudit is weakened to allow encoding by more general quantum operators. We then describe a dense-coding protocol that can overcome this limitation with high probability, assuming the largest Schmidt coefficient of $|\ensuremath{\Psi}\ensuremath{\rangle}$ is sufficiently close to 1/d. In this protocol, ${d}^{2}\ensuremath{-}2$ of the messages are encoded via unitary operations on Alice's qudit, and the final $({d}^{2}\ensuremath{-}1)$-th message is encoded via a non-trace-preserving quantum operation.
We consider dense coding with partially entangled states on bipartite systems of dimension $d\ifmmode\times\else\texttimes\fi{}d$, studying the conditions under which a given number of messages, N, can be deterministically transmitted. It is known that the largest Schmidt coefficient, ${\ensuremath{\lambda}}_{0}$, must obey the bound ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/N$, and considerable empirical evidence points to the conclusion that there exist states satisfying ${\ensuremath{\lambda}}_{0}=d/N$ for every d and N except the special cases N=d+1 and $N={d}^{2}\ensuremath{-}1$. We provide additional conditions under which this bound cannot be reached---that is, when it must be that ${\ensuremath{\lambda}}_{0}<d/N$---yielding insight into the shapes of boundaries separating entangled states that allow N messages from those that allow only $N\ensuremath{-}1$. We also show that these conclusions hold no matter what operations are used for the encoding, and in so doing, identify circumstances under which unitary encoding is strictly better than nonunitary.