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arXiv:0809.0843 · Physical Review A

Augmented message-matrix approach to deterministic dense-coding theory

E. Gerjuoy, H. T. Williams, Paul Bourdon

Published Apr 13, 2009• 3 citations
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Abstract

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 $K$ perfectly distinguishable messages is attainable via $K$ selected local unitary operations performed on one qudit from a pair of entangled qudits of equal dimension $d$ in a pure state $|\ensuremath{\psi}⟩$ with largest Schmidt coefficient $\sqrt{{\ensuremath{\lambda}}_{0}}$. The method takes advantage of the fact that the $K$ message states, together with ${d}^{2}\ensuremath{-}K$ augmenting orthonormal state vectors, yield a unitary matrix, thereby implying properties of the $K$ 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 $K=d+1$, and (iii) the impossibility of finding a $|\ensuremath{\psi}⟩$ that can enable transmission of $K={d}^{2}\ensuremath{-}1$ messages but not ${d}^{2}$. 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 $K=d+1$ 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 $m$ non-negative integral powers of the shift operator $X$.

DOI
10.1103/physreva.79.042315

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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 $K=2d\ensuremath{-}1$.
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.
In a seminal work [PRL85.4972], Walgate, Short, Hardy, and Vedral prove in finite dimensions that for every pair of pure multipartite orthogonal quantum states, there exists a one-way local operations and classical communication (LOCC) protocol that perfectly distinguishes the pair. We extend this result to infinite dimensions with a simpler proof. For states on CdA×dA⊗CdB×dB\mathbb{C}^{d_A \times d_A} \otimes \mathbb{C}^{d_B \times d_B}CdA​×dA​⊗CdB​×dB​, we strengthen this existence result by constructing an O(dA2dB2)O(d_A^2 d_B^2)O(dA2​dB2​)-time algorithm that specifies such a perfect one-way LOCC protocol. Finally, we establish the equivalence between Walgate et al.'s result and the fact that the one-shot environment-assisted classical capacity of every quantum channel is at least 1 bit per channel use, thereby clarifying the literature on these notions. At the core of all of these results is the fact that every operator with vanishing trace admits a basis where its diagonal entries are all zero.
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 ${\ensuremath{\tau}}_{x}$) Bob knows for sure that Alice sent message xxx, 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{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 $\overline{D}\ensuremath{\leqslant}D$, assuming ${\ensuremath{\tau}}_{x}>0$ for a set of D‾D\overline{D}DDD 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\overline{D}=DD=D) that are orthogonal for different messages. Both bounds are saturated when ${\ensuremath{\tau}}_{x}$ 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.
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.
I consider the deterministic distinguishability of a set of orthogonal, bipartite states when only a single copy is available and the parties are restricted to local operations and classical communication, but with the additional requirement that entanglement must be preserved in the process. Several general theorems aimed at characterizing sets of states with which the parties can succeed in such a task are proven. These include (i) a maximum for the number of states when the Schmidt rank of every outcome must be at least a given minimum, (ii) an upper bound (equal to the dimension of Hilbert space if entanglement need not be preserved) for the sum over Schmidt ranks of the initial states when only one-way classical communication is allowed, and (iii) separately, a necessary and a sufficient condition on the states such that their original Schmidt ranks can always be preserved. Two additional theorems explicitly demonstrate a trade-off between the extent to which the set of states fill Hilbert space, as measured by their Schmidt ranks, and how refined the parties must make their measurements, an important factor in determining the Schmidt rank the state can retain after it has been identified. It is shown that our bound on the sum of Schmidt ranks can be exceeded if two-way communication is permitted, and this includes the case that entanglement need not be preserved, so that this sum can exceed the dimension of Hilbert space. Such questions, concerning how the various results are affected by the resources used by the parties (amount of classical communication and types of local operations), are addressed for each theorem. This subject is closely related to the problem of locally purifying an entangled state from a mixed state, which is of direct relevance to teleportation and dense coding using a mixed-state resource. In an appendix, I give an extremely simple and transparent proof of ``nonlocality without entanglement,'' a phenomenon originally discussed by Bennett and co-workers several years ago.