Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0906.0384 · Physical Review A

Overcoming a limitation of deterministic dense coding with a nonmaximally entangled initial state

Paul Bourdon, E. Gerjuoy

Published Feb 16, 2010• 4 citations
arXivHTMLPDF

Abstract

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 $\sqrt{1/d}$), then Alice can convey to Bob one of ${d}^{2}$ 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 $\sqrt{1/d}$. 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.

DOI
10.1103/physreva.81.022314

Related papers

Deterministic dense coding and entanglement entropy

Paul Bourdon, E. Gerjuoy, J. P. McDonald +1 more

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 $K=2d\ensuremath{-}1$.

arXiv:0709.0563•Physical Review A•Feb 5, 2008•20 citations

Deterministic and unambiguous dense coding

Shengjun Wu, Scott M. Cohen, Yuqing Sun +1 more

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.

arXiv:quant-ph/0512169•Physical Review A•Apr 11, 2006•40 citations

Augmented message-matrix approach to deterministic dense-coding theory

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

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 $X$.

arXiv:0809.0843•Physical Review A•Apr 13, 2009•3 citations

Phase boundaries in deterministic dense coding

Michael R. Beran, Scott M. Cohen

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.

arXiv:0810.3608•Physical Review A•Mar 6, 2009•7 citations

Nonoptimality of unitary encoding with quantum channels assisted by entanglement

Michael R. Beran, Scott M. Cohen

One of the primary goals of information theory is to provide limits on the amount of information one can send through communication channels, and to understand encoding methods that achieve such limits. The discovery of dense coding showed that quantum channels can transmit twice as much classical information than is possible with a classical channel. This optimal rate requires using quantum systems maximally entangled between sender and receiver, and then the sender can encode her messages using unitary operations. The use of nonmaximally entangled systems has been intensively studied in recent years, and it has appeared as though unitary encoding might well be optimal in all cases. Indeed, this optimality of unitary operations has been found to hold under far more general conditions. Nonetheless, we here present numerical evidence supported by analytical arguments that there exist circumstances where more classical information can be encoded using nonunitary, rather than unitary, operations.

arXiv:0807.4552•Physical Review A•Dec 23, 2008•10 citations

Deterministic dense coding with partially entangled states

Shay Mozes, Jonathan Oppenheim, Benni Reznik

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 $[d,{d}^{2}]$ 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.

arXiv:quant-ph/0403189•Physical Review A•Jan 7, 2005•74 citations
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 ∣\ensuremathΨ\ensuremath⟩|\ensuremath{\Psi}\ensuremath{\rangle}∣\ensuremathΨ\ensuremath⟩. If ∣\ensuremathΨ\ensuremath⟩|\ensuremath{\Psi}\ensuremath{\rangle}∣\ensuremathΨ\ensuremath⟩ 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 ∣\ensuremathΨ\ensuremath⟩|\ensuremath{\Psi}\ensuremath{\rangle}∣\ensuremathΨ\ensuremath⟩ 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 d2\ensuremath−1{d}^{2}\ensuremath{-}1d2\ensuremath−1 messages. Encoding d2\ensuremath−2{d}^{2}\ensuremath{-}2d2\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 ∣\ensuremathΨ\ensuremath⟩|\ensuremath{\Psi}\ensuremath{\rangle}∣\ensuremathΨ\ensuremath⟩ is not maximally entangled, the communications limit of d2\ensuremath−2{d}^{2}\ensuremath{-}2d2\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 ∣\ensuremathΨ\ensuremath⟩|\ensuremath{\Psi}\ensuremath{\rangle}∣\ensuremathΨ\ensuremath⟩ is sufficiently close to 1/d\sqrt{1/d}1/d​. In this protocol, d2\ensuremath−2{d}^{2}\ensuremath{-}2d2\ensuremath−2 of the messages are encoded via unitary operations on Alice's qudit, and the final (d2\ensuremath−1)({d}^{2}\ensuremath{-}1)(d2\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 $K=2d\ensuremath{-}1$.
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.
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.
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.
One of the primary goals of information theory is to provide limits on the amount of information one can send through communication channels, and to understand encoding methods that achieve such limits. The discovery of dense coding showed that quantum channels can transmit twice as much classical information than is possible with a classical channel. This optimal rate requires using quantum systems maximally entangled between sender and receiver, and then the sender can encode her messages using unitary operations. The use of nonmaximally entangled systems has been intensively studied in recent years, and it has appeared as though unitary encoding might well be optimal in all cases. Indeed, this optimality of unitary operations has been found to hold under far more general conditions. Nonetheless, we here present numerical evidence supported by analytical arguments that there exist circumstances where more classical information can be encoded using nonunitary, rather than unitary, operations.
The utilization of a ddd-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>2d>2d>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 ddd and 2d2d2d. In general, we numerically find that the maximal alphabet size is any integer in the range [d,d2][d,{d}^{2}][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.