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arXiv:quant-ph/0512169 · Physical Review A

Deterministic and unambiguous dense coding

Shengjun Wu, Scott M. Cohen, Yuqing Sun, Robert B. Griffiths

Published Apr 11, 2006• 40 citations
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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.

DOI
10.1103/physreva.73.042311

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Optimal dense coding using a partially-entangled pure state of Schmidt rank D‾\overline{D}D and a noiseless quantum channel of dimension DDD is studied both in the deterministic case where at most Ld{L}_{d}Ld​ messages can be transmitted with perfect fidelity, and in the unambiguous case where when the protocol succeeds (probability \ensuremathτx{\ensuremath{\tau}}_{x}\ensuremathτx​) Bob knows for sure that Alice sent message xxx, and when it fails (probability 1\ensuremath−\ensuremathτx1\ensuremath{-}{\ensuremath{\tau}}_{x}1\ensuremath−\ensuremathτx​) he knows it has failed. Alice is allowed any single-shot (one use) encoding procedure, and Bob any single-shot measurement. For D‾\ensuremath⩽D\overline{D}\ensuremath{\leqslant}DD\ensuremath⩽D a bound is obtained for Ld{L}_{d}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\overline{D}>DD>D it is shown that Ld{L}_{d}Ld​ is strictly less than D2{D}^{2}D2 unless D‾\overline{D}D is an integer multiple of DDD, in which case uniform (maximal) entanglement is not needed to achieve the optimal protocol. The unambiguous case is studied for D‾\ensuremath⩽D\overline{D}\ensuremath{\leqslant}DD\ensuremath⩽D, assuming \ensuremathτx>0{\ensuremath{\tau}}_{x}>0\ensuremathτx​>0 for a set of D‾D\overline{D}DDD messages, and a bound is obtained for the average ⟨1∕\ensuremathτ⟩⟨1∕\ensuremath{\tau}⟩⟨1∕\ensuremathτ⟩. A bound on the average ⟨\ensuremathτ⟩⟨\ensuremath{\tau}⟩⟨\ensuremathτ⟩ requires an additional assumption of encoding by isometries (unitaries when D‾=D\overline{D}=DD=D) that are orthogonal for different messages. Both bounds are saturated when \ensuremathτx{\ensuremath{\tau}}_{x}\ensuremathτx​ is a constant independent of xxx, by a protocol based on one-shot entanglement concentration. For D‾>D\overline{D}>DD>D it is shown that (at least) D2{D}^{2}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\sqrt{1/d}1/d​), then Alice can convey to Bob one of d2{d}^{2}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\sqrt{1/d}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 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=3d=3d=3 and d=4d=4d=4, a conjecture of Mozes et al. about the minimum entanglement entropy that supports the sending of d+jd+jd+j messages, $2\ensuremath{\le}j\ensuremath{\le}d\ensuremath{-}1$; moreover, we show that the j=2j=2j=2 and $j=d\ensuremath{-}1$ cases of the conjecture are valid in all dimensions. (iii) Given that $|\ensuremath{\psi}⟩$ allows the sending of KKK 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+1K=d+1K=d+1, while our constructions of local unitaries show that equality can be realized if K=d+2K=d+2K=d+2 or $K=2d\ensuremath{-}1$.
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, NNN, 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 ddd and NNN except the special cases N=d+1N=d+1N=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 NNN 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.
A useful method for deriving analytical results applicable to the standard two-party deterministic dense-coding protocol is introduced and illustrated. In this protocol, communication of KKK perfectly distinguishable messages is attainable via KKK selected local unitary operations performed on one qudit from a pair of entangled qudits of equal dimension ddd in a pure state $|\ensuremath{\psi}⟩$ with largest Schmidt coefficient $\sqrt{{\ensuremath{\lambda}}_{0}}$. The method takes advantage of the fact that the KKK message states, together with ${d}^{2}\ensuremath{-}K$ augmenting orthonormal state vectors, yield a unitary matrix, thereby implying properties of the KKK message states which otherwise are not readily recognized. Employing this augmented message matrix, we produce simple proofs of previously established results including (i) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$, (ii) ${\ensuremath{\lambda}}_{0}<d/K$ when K=d+1K=d+1K=d+1, and (iii) the impossibility of finding a $|\ensuremath{\psi}⟩$ that can enable transmission of $K={d}^{2}\ensuremath{-}1$ messages but not d2{d}^{2}d2. Additional results obtained using the method include proofs that the ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$ bound is reduced to at least (i) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}(1/2)[1+\sqrt{(d\ensuremath{-}2)/(d+2)}]$ when K=d+1K=d+1K=d+1 and (ii) ${\ensuremath{\lambda}}_{0}\ensuremath{\le}(K\ensuremath{-}m)/(2K\ensuremath{-}m\ensuremath{-}d)$ whenever $(d+1)\ensuremath{\le}K\ensuremath{\le}2d$ and the selected local unitaries include the first mmm non-negative integral powers of the shift operator XXX.
Unambiguous unitary maps and unambiguous unitary quantum channels are introduced and some of their properties are derived. These properties ensures certain simple form for the measurements involved in realizing an unambiguous unitary quantum channel. Error correction and unambiguous error correction with nonzero probability are discussed in terms of unambiguous unitary quantum channels. We not only re-derive the well-known condition for a set of errors to be correctable with certainty, but also obtain a necessary and sufficient condition for the errors caused by a noisy channel to be correctable with any nonzero probability. Dense coding with a partially entangled state can also be viewed as an unambiguous unitary quantum channel when all messages are required to be transmitted with equal probability of success, the maximal achievable probability of success is derived and the optimum protocol is also obtained.